Graph Theory, an Antiprism Graph Is a Graph That Has One of the Antiprisms As Its Skeleton

Home , Antiprism graph, Apollonian network, Arboricity, Arrangement of lines, Asymmetric graph, Audrey Terras

Graph families From Wikipedia, the free encyclopedia Chapter 1

Antiprism graph

In the mathematical ﬁeld of graph theory, an antiprism graph is a graph that has one of the antiprisms as its skeleton. An n-sided antiprism has 2n vertices and 4n edges. They are regular, polyhedral (and therefore by necessity also 3- vertex-connected, vertex-transitive, and planar graphs), and also Hamiltonian graphs.[1]

1.1 Examples

The ﬁrst graph in the sequence, the octahedral graph, has 6 vertices and 12 edges. Later graphs in the sequence may be named after the type of antiprism they correspond to:

• Octahedral graph – 6 vertices, 12 edges

• square antiprismatic graph – 8 vertices, 16 edges

• Pentagonal antiprismatic graph – 10 vertices, 20 edges

• Hexagonal antiprismatic graph – 12 vertices, 24 edges

• Heptagonal antiprismatic graph – 14 vertices, 28 edges

• Octagonal antiprismatic graph– 16 vertices, 32 edges

• ...

Although geometrically the star polygons also form the faces of a diﬀerent sequence of (self-intersecting) antiprisms, the star antiprisms, they do not form a diﬀerent sequence of graphs.

1.2 Related graphs

An antiprism graph is a special case of a circulant graph, Ci₂n(2,1). Other inﬁnite sequences of polyhedral graph formed in a similar way from polyhedra with regular-polygon bases include the prism graphs (graphs of prisms) and wheel graphs (graphs of pyramids). Other vertex-transitive polyhedral graphs include the Archimedean graphs.

1.3 References

[1] Read, R. C. and Wilson, R. J. An Atlas of Graphs, Oxford, England: Oxford University Press, 2004 reprint, Chapter 6 special graphs pp. 261, 270.

2 1.4. EXTERNAL LINKS 3

1.4 External links

• Weisstein, Eric W., “Antiprism graph”, MathWorld. Chapter 2

Aperiodic graph

In the mathematical area of graph theory, a directed graph is said to be aperiodic if there is no integer k > 1 that divides the length of every cycle of the graph. Equivalently, a graph is aperiodic if the greatest common divisor of the lengths of its cycles is one; this greatest common divisor for a graph G is called the period of G.

2.1 Graphs that cannot be aperiodic

In any directed bipartite graph, all cycles have a length that is divisible by two. Therefore, no directed bipartite graph can be aperiodic. In any directed acyclic graph, it is a vacuous truth that every k divides all cycles (because there are no directed cycles to divide) so no directed acyclic graph can be aperiodic. And in any directed cycle graph, there is only one cycle, so every cycle’s length is divisible by n, the length of that cycle.

2.2 Testing for aperiodicity

Suppose that G is strongly connected and that k divides the lengths of all cycles in G. Consider the results of performing a depth-first search of G, starting at any vertex, and assigning each vertex v to a set Vi where i is the length (taken mod k) of the path in the depth-first search tree from the root to v. It can be shown (Jarvis & Shier 1996) that this partition into sets Vi has the property that each edge in the graph goes from a set Vi to another set V₍i ₊ ₁₎ ₒ k. Conversely, if a partition with this property exists for a strongly connected graph G, k must divide the lengths of all cycles in G. Thus, we may find the period of a strongly connected graph G by the following steps:

• Perform a depth-ﬁrst search of G • For each e in G that connects a vertex on level i of the depth-ﬁrst search tree to a vertex on level j, let ke = j - i - 1. • Compute the greatest common divisor of the set of numbers ke.

The graph is aperiodic if and only if the period computed in this fashion is 1. If G is not strongly connected, we may perform a similar computation in each strongly connected component of G, ignoring the edges that pass from one strongly connected component to another. Jarvis and Shier describe a very similar algorithm using breadth first search in place of depth-first search; the advantage of depth-first search is that the strong connectivity analysis can be incorporated into the same search.

2.3 Applications

In a strongly connected graph, if one deﬁnes a Markov chain on the vertices, in which the probability of transitioning from v to w is nonzero if and only if there is an edge from v to w, then this chain is aperiodic if and only if the graph

4 2.3. APPLICATIONS 5

An aperiodic graph. The cycles in this graph have lengths 5 and 6; therefore, there is no k > 1 that divides all cycle lengths. 6 CHAPTER 2. APERIODIC GRAPH

A strongly connected graph with period three. is aperiodic. A Markov chain in which all states are recurrent has a strongly connected state transition graph, and the Markov chain is aperiodic if and only if this graph is aperiodic. Thus, aperiodicity of graphs is a useful concept in analyzing the aperiodicity of Markov chains. Aperiodicity is also an important necessary condition for solving the road coloring problem. According to the solution of this problem (Trahtman 2009), a strongly connected directed graph in which all vertices have the same outdegree has a synchronizable edge coloring if and only if it is aperiodic.

2.4 References

• Jarvis, J. P.; Shier, D. R. (1996), “Graph-theoretic analysis of ﬁnite Markov chains”, in Shier, D. R.; Wallenius, K. T., Applied Mathematical Modeling: A Multidisciplinary Approach (PDF), CRC Press.

• Trahtman, Avraham N. (2009), “The road coloring problem”, Israel Journal of Mathematics 172 (1): 51–60, arXiv:0709.0099, doi:10.1007/s11856-009-0062-5. Chapter 3

Apex graph

An apex graph. The subgraph formed by removing the red vertex is planar.

7 8 CHAPTER 3. APEX GRAPH

In graph theory, a branch of mathematics, an apex graph is a graph that can be made planar by the removal of a single vertex. The deleted vertex is called an apex of the graph. We say an apex, not the apex because an apex graph may have more than one apex (for example, in the minimal nonplanar graphs K5 or K₃,₃, every vertex is an apex). The apex graphs include graphs that are themselves planar, in which case again every vertex is an apex. The null graph is also counted as an apex graph even though it has no vertex to remove. Apex graphs are closed under the operation of taking minors and play a role in several other aspects of graph minor theory: linkless embedding,[1] Hadwiger’s conjecture,[2] YΔY-reducible graphs,[3] and relations between treewidth and graph diameter.[4]

3.1 Characterization and recognition

Apex graphs are closed under the operation of taking minors: contracting any edge, or removing any edge or vertex, leads to another apex graph. For, if G is an apex graph with apex v, then any contraction or removal that does not involve v preserves the planarity of the remaining graph, as does any edge removal of an edge incident to v. If an edge incident to v is contracted, the effect on the remaining graph is equivalent to the removal of the other endpoint of the edge. And if v itself is removed, any other vertex may be chosen as the apex.[5] Because they form a minor-closed family of graphs, the apex graphs have a forbidden graph characterization: there exists a finite set A of minor-minimal non-apex graphs such that a graph is an apex graph if and only if it does not contain as a minor any graph in A. The forbidden minors for the apex graphs include the seven graphs of the Petersen family, three disconnected graphs formed from the disjoint unions of two of K5 and K₃,₃, and many other graphs. However, a complete description of the graphs in A remains unknown.[5][6] Despite the unknown set of forbidden minors, it is possible to test whether a given graph is an apex graph, and if so, to find an apex for the graph, in linear time. More generally, for any fixed constant k, it is possible to recognize in linear time the k-apex graphs, the graphs in which the removal of some carefully chosen set of at most k vertices leads to a planar graph.[7] If k is variable, however, the problem is NP-complete.[8]

3.2 Chromatic number

Every apex graph has chromatic number at most ﬁve: the underlying planar graph requires at most four colors by the four color theorem, and the remaining vertex needs at most one additional color. Robertson, Seymour & Thomas (1993a) used this fact in their proof of the case k = 6 of the Hadwiger conjecture, the statement that every 6-chromatic graph has the complete graph K6 as a minor: they showed that any minimal counterexample to the conjecture would have to be an apex graph, but since there are no 6-chromatic apex graphs such a counterexample cannot exist.

Jørgensen (1994) conjectured that every 6-vertex-connected graph that does not have K6 as a minor must be an apex graph. If this were proved, the Robertson–Seymour–Thomas result on the Hadwiger conjecture would be an immediate consequence.[2]

3.3 Local treewidth

A graph family F has bounded local treewidth if the graphs in F obey a functional relationship between diameter and treewidth: there exists a function ƒ such that the treewidth of a diameter-d graph in F is at most ƒ(d). The apex graphs do not have bounded local treewidth: the apex graphs formed by connecting an apex vertex to every vertex of an n × n grid graph have treewidth n and diameter 2, so the treewidth is not bounded by a function of diameter for these graphs. However, apex graphs are intimately connected to bounded local treewidth: the minor-closed graph families F that have bounded local treewidth are exactly the families that have an apex graph as one of their forbidden minors.[4] A minor-closed family of graphs that has an apex graph as one of its forbidden minors is known as apex- minor-free. With this terminology, the connection between apex graphs and local treewidth can be restated as the fact that apex-minor-free graph families are the same as minor-closed graph families with bounded local treewidth. The concept of bounded local treewidth forms the basis of the theory of bidimensionality, and allows for many algorithmic problems on apex-minor-free graphs to be solved exactly by a polynomial-time algorithm or a ﬁxed-parameter tractable algorithm, or approximated using a polynomial-time approximation scheme.[9] Apex-minor-free graph families obey a strengthened version of the graph structure theorem, leading to additional approximation algorithms for 3.4. EMBEDDINGS 9

graph coloring and the travelling salesman problem.[10] However, some of these results can also be extended to arbitrary minor-closed graph families via structure theorems relating them to apex-minor-free graphs.[11]

3.4 Embeddings

If G is an apex graph with apex v, and τ is the minimum number of faces needed to cover all the neighbors of v in a planar embedding of G\{v}, then G may be embedded onto a two-dimensional surface of genus τ − 1: simply add that number of bridges to the planar embedding, connecting together all the faces into which v must be connected. For instance, adding a single vertex to an outerplanar graph (a graph with τ = 1) produces a planar graph. When G\{v} is 3-connected, his bound is within a constant factor of optimal: every surface embedding of G requires genus at least τ/160. However, it is NP-hard to determine the optimal genus of a surface embedding of an apex graph.[12] By using SPQR trees to encode the possible embeddings of the planar part of an apex graph, it is possible to compute a drawing of the graph in the plane in which the only crossings involve the apex vertex, minimizing the total number of crossings, in polynomial time.[13] However, if arbitrary crossings are allowed, it becomes NP-hard to minimize the number of crossings, even in the special case of apex graphs formed by adding a single edge to a planar graph.[14] Apex graphs are also linklessly embeddable in three-dimensional space: they can be embedded in such a way that each cycle in the graph is the boundary of a disk that is not crossed by any other feature of the graph.[15] A drawing of this type may be obtained by drawing the planar part of the graph in a plane, placing the apex above the plane, and connecting the apex by straight-line edges to each of its neighbors. Linklessly embeddable graphs form a minor-closed family with the seven graphs in the Petersen family as their minimal forbidden minors;[1] therefore, these graphs are also forbidden as minors for the apex graphs. However, there exist linklessly embeddable graphs that are not apex graphs.

3.5 YΔY-reducibility

A connected graph is YΔY-reducible if it can be reduced to a single vertex by a sequence of steps, each of which is a Δ-Y or Y-Δ transform, the removal of a self-loop or multiple adjacency, the removal of a vertex with one neighbor, and the replacement of a vertex of degree two and its two neighboring edges by a single edge.[3] Like the apex graphs and the linkless embeddable graphs, the YΔY-reducible graphs are closed under graph minors. And, like the linkless embeddable graphs, the YΔY-reducible graphs have the seven graphs in the Petersen family as forbidden minors, prompting the question of whether these are the only forbidden minors and whether the YΔY- reducible graphs are the same as the linkless embeddable graphs. However, Neil Robertson provided an example of an apex graph that is not YΔY-reducible. Since every apex graph is linkless embeddable, this shows that there are graphs that are linkless embeddable but not YΔY-reducible and therefore that there are additional forbidden minors for the YΔY-reducible graphs.[3] Robertson’s apex graph is shown in the ﬁgure. It can be obtained by connecting an apex vertex to each of the degree- three vertices of a rhombic dodecahedron, or by merging two diametrally opposed vertices of a four-dimensional hypercube graph. Because the rhombic dodecahedron’s graph is planar, Robertson’s graph is an apex graph. It is a triangle-free graph with minimum degree four, so it cannot be changed by any YΔY-reduction.[3]

3.6 Nearly planar graphs

If a graph is an apex graph, it is not necessarily the case that it has a unique apex. For instance, in the minor-minimal nonplanar graphs K5 and K₃,₃, any of the vertices can be chosen as the apex. Wagner (1967, 1970) defined a nearly planar graph to be a nonplanar apex graph with the property that all vertices can be the apex of the graph; thus, K5 and K₃,₃ are nearly planar. He provided a classification of these graphs into four subsets, one of which consists of the graphs that (like the Möbius ladders) can be embedded onto the Möbius strip in such a way that the single edge of the strip coincides with a Hamiltonian cycle of the graph. Prior to the proof of the four color theorem, he proved that every nearly planar graph can be colored with at most four colors, except for the graphs formed from a wheel graph with an odd outer cycle by replacing the hub vertex with two adjacent vertices, which require five colors. Additionally, he proved that, with a single exception (the eight-vertex complement graph of the cube) every nearly planar graph has an embedding onto the projective plane. 10 CHAPTER 3. APEX GRAPH

Robertson’s example of a non-YΔY-reducible apex graph.

However, the phrase “nearly planar graph” is highly ambiguous: it has also been used to refer to apex graphs,[16] graphs formed by adding one edge to a planar graph,[17] and graphs formed from a planar embedded graph by replacing a bounded number of faces by “vortexes” of bounded pathwidth,[18] as well as for other less precisely-deﬁned sets of graphs.

3.7 See also

• Polyhedral pyramid, a 4-dimensional polytope whose vertices and edges form an apex graph, with the apex adjacent to every vertex of a polyhedral graph

3.8 Notes

[1] Robertson, Seymour & Thomas (1993b).

[2] Robertson, Seymour & Thomas (1993a).

[3] Truemper (1992). 3.9. REFERENCES 11

The 16-vertex Möbius ladder, an example of a nearly planar graph.

[4] Eppstein (2000); Demaine & Hajiaghayi (2004).

[5] Gupta & Impagliazzo (1991).

[6] Pierce (2014).

[7] Kawarabayashi (2009).

[8] Lewis & Yannakakis (1980).

[9] Eppstein (2000); Frick & Grohe (2001); Demaine & Hajiaghayi (2005).

[10] Demaine, Hajiaghayi & Kawarabayashi (2009).

[11] Grohe (2003).

[12] Mohar (2001).

[13] Chimani et al. (2009).

[14] Cabello & Mohar (2010).

[15] Robertson, Seymour & Thomas (1993c).

[16] Robertson, Seymour & Thomas (1993c); Eppstein (2000).

[17] Archdeacon & Bonnington (2004).

[18] Abraham & Gavoille (2006).

3.9 References

• Abraham, Ittai; Gavoille, Cyril (2006), “Object location using path separators”, Proc. 25th ACM Symposium on Principles of Distributed Computing (PODC '06), pp. 188–197, doi:10.1145/1146381.1146411.

• Archdeacon, Dan; Bonnington, C.P.C. Paul (2004), “Obstructions for embedding cubic graphs on the spindle surface”, Journal of Combinatorial Theory, Series B 91 (2): 229–252, doi:10.1016/j.jctb.2004.02.001.

• Cabello, Sergio; Mohar, Bojan (2010), “Adding one edge to planar graphs makes crossing number hard”, Proc. 26th ACM Symposium on Computational Geometry (SoCG '10) (PDF), pp. 68–76, doi:10.1145/1810959.1810972.

• Chimani, Markus; Gutwenger, Carsten; Mutzel, Petra; Wolf, Christian (2009), “Inserting a vertex into a planar graph”, Proc. 20th ACM-SIAM Symposium on Discrete Algorithms (SODA '09), pp. 375–383. 12 CHAPTER 3. APEX GRAPH

• Demaine, Erik D.; Hajiaghayi, Mohammad Taghi (2004), “Diameter and treewidth in minor-closed graph families, revisited”, Algorithmica 40 (3): 211–215, doi:10.1007/s00453-004-1106-1. • Demaine, Erik D.; Hajiaghayi, Mohammad Taghi (2005), “Bidimensionality: new connections between FPT algorithms and PTASs”, Proc. 16th ACM-SIAM Symposium on Discrete Algorithms (SODA '05), pp. 590–601. • Demaine, Erik D.; Hajiaghayi, Mohammad Taghi; Kawarabayashi, Ken-ichi (2009), “Approximation algorithms via structural results for apex-minor-free graphs” (PDF), Proc. 36th International Colloquium Automata, Languages and Programming (ICALP '09), Lecture Notes in Computer Science 5555, Springer-Verlag, pp. 316–327, doi:10.1007/978-3-642-02927-1_27. • Eppstein, David (2000), “Diameter and treewidth in minor-closed graph families”, Algorithmica 27 (3): 275– 291, arXiv:math.CO/9907126, doi:10.1007/s004530010020. • Frick, Markus; Grohe, Martin (2001), “Deciding ﬁrst-order properties of locally tree-decomposable structures”, Journal of the ACM 48 (6): 1184–1206, arXiv:cs/0004007, doi:10.1145/504794.504798.

• Grohe, Martin (2003), “Local tree-width, excluded minors, and approximation algorithms”, Combinatorica 23 (4): 613–632, arXiv:math.CO/0001128, doi:10.1007/s00493-003-0037-9.

• Gupta, A.; Impagliazzo, R. (1991), “Computing planar intertwines”, Proc. 32nd IEEE Symposium on Founda- tions of Computer Science (FOCS '91), IEEE Computer Society, pp. 802–811, doi:10.1109/SFCS.1991.185452.

• Jørgensen, Leif K. (1994), “Contractions to K8", Journal of Graph Theory 18 (5): 431–448, doi:10.1002/jgt.3190180502. As cited by Robertson, Seymour, and Thomas (1993a, 1993c).

• Kawarabayashi, Ken-ichi (2009), “Planarity allowing few error vertices in linear time” (PDF), Proc. 50th IEEE Symposium on Foundations of Computer Science (FOCS '09), IEEE Computer Society, pp. 639–648, doi:10.1109/FOCS.2009.45. • Lewis, John M.; Yannakakis, Mihalis (1980), “The node-deletion problem for hereditary properties is NP- complete”, Journal of Computer and System Sciences 20 (2): 219–230, doi:10.1016/0022-0000(80)90060-4. • Mohar, Bojan (2001), “Face covers and the genus problem for apex graphs” (PDF), Journal of Combinatorial Theory, Series B 82 (1): 102–117, doi:10.1006/jctb.2000.2026. • Pierce, Mike (2014), Searching for and classifying the ﬁnite set of minor-minimal non-apex graphs (PDF), Honours thesis, California State University, Chico.

• Robertson, Neil; Seymour, Paul; Thomas, Robin (1993a), “Hadwiger’s conjecture for K6-free graphs” (PDF), Combinatorica 13 (3): 279–361, doi:10.1007/BF01202354. • Robertson, Neil; Seymour, P. D.; Thomas, Robin (1993b), “Linkless embeddings of graphs in 3-space”, Bulletin of the American Mathematical Society 28 (1): 84–89, arXiv:math/9301216, doi:10.1090/S0273-0979- 1993-00335-5, MR 1164063.

• Robertson, Neil; Seymour, Paul; Thomas, Robin (1993c), “A survey of linkless embeddings”, in Robertson, Neil; Seymour, Paul, Graph Structure Theory: Proc. AMS–IMS–SIAM Joint Summer Research Conference on Graph Minors (PDF), Contemporary Mathematics 147, American Mathematical Society, pp. 125–136.

• Truemper, Klaus (1992), Matroid Decomposition (PDF), Academic Press, pp. 100–101. • Wagner, Klaus (1967), “Fastplättbare Graphen”, Journal of Combinatorial Theory (in German) 3 (4): 326– 365, doi:10.1016/S0021-9800(67)80103-0. • Wagner, Klaus (1970), “Zum basisproblem der nicht in die projektive ebene einbettbaren graphen, I”, Journal of Combinatorial Theory (in German) 9 (1): 27–43, doi:10.1016/S0021-9800(70)80052-7. Chapter 4

Apollonian network

An Apollonian network

In combinatorial mathematics, an Apollonian network is an undirected graph formed by a process of recursively subdividing a triangle into three smaller triangles. Apollonian networks may equivalently be deﬁned as the planar 3- trees, the maximal planar chordal graphs, the uniquely 4-colorable planar graphs, and the graphs of stacked polytopes. They are named after Apollonius of Perga, who studied a related circle-packing construction.

13 14 CHAPTER 4. APOLLONIAN NETWORK

The Goldner–Harary graph, a non-Hamiltonian Apollonian network

4.1 Deﬁnition

An Apollonian network may be formed, starting from a single triangle embedded in the Euclidean plane, by repeatedly selecting a triangular face of the embedding, adding a new vertex inside the face, and connecting the new vertex to each vertex of the face containing it. In this way, the triangle containing the new vertex is subdivided into three smaller triangles, which may in turn be subdivided in the same way.

4.2 Examples

The complete graphs on three and four vertices, K3 and K4, are both Apollonian networks. K3 is formed by starting with a triangle and not performing any subdivisions, while K4 is formed by making a single subdivision before stopping. The Goldner–Harary graph is an Apollonian network that forms the smallest non-Hamiltonian maximal planar graph.[1] Another more complicated Apollonian network was used by Nishizeki (1980) to provide an example of a 1-tough non-Hamiltonian maximal planar graph.

4.3 Graph-theoretic characterizations

As well as being deﬁned by recursive subdivision of triangles, the Apollonian networks have several other equivalent mathematical characterizations. They are the chordal maximal planar graphs, the chordal polyhedral graphs, and the planar 3-trees. They are the uniquely 4-colorable planar graphs, and the planar graphs with a unique Schnyder wood decomposition into three trees. They are the maximal planar graphs with treewidth three, a class of graphs that can be characterized by their forbidden minors or by their reducability under Y-Δ transforms. They are the maximal planar graphs with degeneracy three. They are also the planar graphs on a given number of vertices that have the largest possible number of triangles, the largest possible number of tetrahedral subgraphs, the largest possible number of cliques, and the largest possible number of pieces after decomposing by separating triangles. 4.3. GRAPH-THEORETIC CHARACTERIZATIONS 15

4.3.1 Chordality

Apollonian networks are examples of maximal planar graphs, graphs to which no additional edges can be added without destroying planarity, or equivalently graphs that can be drawn in the plane so that every face (including the outer face) is a triangle. They are also chordal graphs, graphs in which every cycle of four or more vertices has a diagonal edge connecting two non-consecutive cycle vertices, and the order in which vertices are added in the subdivision process that forms an Apollonian network is an elimination ordering as a chordal graph. This forms an alternative characterization of the Apollonian networks: they are exactly the chordal maximal planar graphs or equivalently the chordal polyhedral graphs.[2] In an Apollonian network, every maximal clique is a complete graph on four vertices, formed by choosing any vertex and its three earlier neighbors. Every minimal clique separator (a clique that partitions the graph into two disconnected subgraphs) is one of the subdivided triangles. A chordal graph in which all maximal cliques and all minimal clique separators have the same size is a k-tree, and Apollonian networks are examples of 3-trees. Not every 3-tree is planar, but the planar 3-trees are exactly the Apollonian networks.

4.3.2 Unique colorability

Every Apollonian network is also a uniquely 4-colorable graph. Because it is a planar graph, the four color theorem implies that it has a graph coloring with only four colors, but once the three colors of the initial triangle are selected, there is only one possible choice for the color of each successive vertex, so up to permutation of the set of colors it has exactly one 4-coloring. It is more diﬃcult to prove, but also true, that every uniquely 4-colorable planar graph is an Apollonian network. Therefore, Apollonian networks may also be characterized as the uniquely 4-colorable planar graphs.[3] Apollonian networks also provide examples of planar graphs having as few k-colorings as possible for k > 4.[4] The Apollonian networks are also exactly the maximal planar graphs that (once an exterior face is ﬁxed) have a unique Schnyder wood, a partition of the edges of the graph into three interleaved trees rooted at the three vertices of the exterior face.[5]

4.3.3 Treewidth

The Apollonian networks do not form a family of graphs that is closed under the operation of taking graph minors, as removing edges but not vertices from an Apollonian network produces a graph that is not an Apollonian network. However, the planar partial 3-trees, subgraphs of Apollonian networks, are minor-closed. Therefore, according to the Robertson–Seymour theorem, they can be characterized by a ﬁnite number of forbidden minors. The minimal forbidden minors for the planar partial 3-trees are the four minimal graphs among the forbidden minors for the planar graphs and the partial 3-trees: the complete graph K5, the complete bipartite graph K₃,₃, the graph of the octahedron, and the graph of the pentagonal prism. The Apollonian graphs are the maximal graphs that do not have any of these four graphs as a minor.[6] A Y-Δ transform, an operation that replaces a degree-three vertex in a graph by a triangle connecting its neighbors, is suﬃcient (together with the removal of parallel edges) to reduce any Apollonian network to a single triangle, and more generally the planar graphs that can be reduced to a single edge by Y-Δ transforms, removal of parallel edges, removal of degree-one vertices, and compression of degree-two vertices are exactly the planar partial 3-trees. The dual graphs of the planar partial 3-trees form another minor-closed graph family and are exactly the planar graphs that can be reduced to a single edge by Δ-Y transforms, removal of parallel edges, removal of degree-one vertices, and compression of degree-two vertices.[7]

4.3.4 Degeneracy

In every subgraph of an Apollonian network, the most recently added vertex has degree at most three, so Apollonian networks have degeneracy three. The order in which the vertices are added to create the network is therefore a degeneracy ordering, and the Apollonian networks coincide with the 3-degenerate maximal planar graphs. 16 CHAPTER 4. APOLLONIAN NETWORK

4.3.5 Extremality

Another characterization of the Apollonian networks involves their connectivity. Any maximal planar graph may be decomposed into 4-vertex-connected maximal planar subgraphs by splitting it along its separating triangles (triangles that are not faces of the graph): given any non-facial triangle: one can form two smaller maximal planar graphs, one consisting of the part inside the triangle and the other consisting of the part outside the triangle. The maximal planar graphs without separating triangles that may be formed by repeated splits of this type are sometimes called blocks, although that name has also been used for the biconnected components of a graph that is not itself biconnected. An Apollonian network is a maximal planar graph in which all of the blocks are isomorphic to the complete graph K4. In extremal graph theory, Apollonian networks are also exactly the n-vertex planar graphs in which the number of blocks achieves its maximum, n − 3, and the planar graphs in which the number of triangles achieves its maximum, 3n − 8. Since each K4 subgraph of a planar graph must be a block, these are also the planar graphs in which the number of K4 subgraphs achieves its maximum, n − 3, and the graphs in which the number of cliques of any type achieves its maximum, 8n − 16.[8]

4.4 Geometric realizations

4.4.1 Construction from circle packings

Apollonian networks are named after Apollonius of Perga, who studied the Problem of Apollonius of constructing a circle tangent to three other circles. One method of constructing Apollonian networks is to start with three mutually- tangent circles and then repeatedly inscribe another circle within the gap formed by three previously-drawn circles. The fractal collection of circles produced in this way is called an Apollonian gasket. If the process of producing an Apollonian gasket is stopped early, with only a ﬁnite set of circles, then the graph that has one vertex for each circle and one edge for each pair of tangent circles is an Apollonian network.[9] The existence of a set of tangent circles whose tangencies represent a given Apollonian network forms a simple instance of the Koebe–Andreev–Thurston circle-packing theorem, which states that any planar graph can be represented by tangent circles in the same way.[10]

4.4.2 Polyhedra

Apollonian networks are planar 3-connected graphs and therefore, by Steinitz’s theorem, can always be represented as the graphs of convex polyhedra. The convex polyhedron representing an Apollonian network is s a 3-dimensional stacked polytope. Such a polytope can be obtained from a tetrahedron by repeatedly gluing additional tetrahedra one at a time onto its triangular faces. Therefore, Apollonian networks may also be deﬁned as the graphs of stacked 3d polytopes.[11] It is possible to ﬁnd a representation of any Apollonian network as convex 3d polyhedron in which all of the coordinates are integers of polynomial size, better than what is known for other planar graphs.[12]

4.4.3 Triangle meshes

The recursive subdivision of triangles into three smaller triangles was investigated as an image segmentation technique in computer vision by Elcock, Gargantini & Walsh (1987); in this context, they called it the ternary scalene triangle decomposition. They observed that, by placing each new vertex at the centroid of its enclosing triangle, the triangulation could be chosen in such a way that all triangles have equal areas, although they do not all have the same shape. More generally, Apollonian networks may be drawn in the plane with any prescribed area in each face; if the areas are rational numbers, so are all of the vertex coordinates.[13] It is also possible to carry out the process of subdividing a triangle to form an Apollonian network in such a way that, at every step, the edge lengths are rational numbers; it is an open problem whether every planar graph has a drawing with this property.[14] It is possible in polynomial time to ﬁnd a drawing of a planar 3-tree with integer coordinates minimizing the area of the bounding box of the drawing, and to test whether a given planar 3-tree may be drawn with its vertices on a given set of points.[15] 4.5. PROPERTIES AND APPLICATIONS 17

An example of an Apollonian gasket

4.5 Properties and applications

4.5.1 Matching-free graphs

Plummer (1992) used Apollonian networks to construct an infinite family of maximal planar graphs with an even number of vertices but with no perfect matching. Plummer’s graphs are formed in two stages. In the first stage, starting from a triangle abc, one repeatedly subdivides the triangular face of the subdivision that contains edge bc: the result is a graph consisting of a path from a to the final subdivision vertex together with an edge from each path vertex to each of b and c. In the second stage, each of the triangular faces of the resulting planar graph is subdivided one more time. If the path from a to the final subdivision vertex of the first stage has even length, then the number of vertices in the overall graph is also even. However, approximately 2/3 of the vertices are the ones inserted in the second stage; these form an independent set, and cannot be matched to each other, nor are there enough vertices outside the independent set to find matches for all of them. Although Apollonian networks themselves may not have perfect matchings, the planar dual graphs of Apollonian networks are 3-regular graphs with no cut edges, so by a theorem of Petersen (1891) they are guaranteed to have at least one perfect matching. However, in this case more is known: the duals of Apollonian networks always have an exponential number of perfect matchings.[16] László Lovász and Michael D. Plummer conjectured that a similar exponential lower bound holds more generally for every 3-regular graph without cut edges, a result that was later 18 CHAPTER 4. APOLLONIAN NETWORK

proven.

4.5.2 Power law graphs

Andrade et al. (2005) studied power laws in the degree sequences of a special case of networks of this type, formed by subdividing all triangles the same number of times. They used these networks to model packings of space by particles of varying sizes. Based on their work, other authors introduced random Apollonian networks, formed by repeatedly choosing a random face to subdivide, and they showed that these also obey power laws in their degree distribution [17] and have small average distances.[18] Alan M. Frieze and Charalampos E. Tsourakakis analyzed the highest degrees and the eigenvalues of random Apollonian networks .[19] Andrade et al. also observed that their networks satisfy the small world eﬀect, that all vertices are within a small distance of each other. Based on numerical evidence they estimated the average distance between randomly selected pairs of vertices in an n-vertex network of this type to be proportional to (log n)3/4, but later researchers showed that the average distance is actually proportional to log n.[20]

4.5.3 Angle distribution

Butler & Graham (2010) observed that if each new vertex is placed at the incenter of its triangle, so that the edges to the new vertex bisect the angles of the triangle, then the set of triples of angles of triangles in the subdivision, when reinterpreted as triples of barycentric coordinates of points in an equilateral triangle, converges in shape to the Sierpinski triangle as the number of levels of subdivision grows.

4.5.4 Hamiltonicity

Takeo (1960) claimed erroneously that all Apollonian networks have Hamiltonian cycles; however, the Goldner– Harary graph provides a counterexample. If an Apollonian network has toughness greater than one (meaning that removing any set of vertices from the graph leaves a smaller number of connected components than the number of removed vertices) then it necessarily has a Hamiltonian cycle, but there exist non-Hamiltonian Apollonian networks whose toughness is equal to one.[21]

4.5.5 Enumeration

The combinatorial enumeration problem of counting Apollonian triangulations was studied by Takeo (1960), who showed that they have the simple generating function f(x) described by the equation f(x) = 1 + x(f(x))3. In this generating function, the term of degree n counts the number of Apollonian networks with a ﬁxed outer triangle and n + 3 vertices. Thus, the numbers of Apollonian networks (with a ﬁxed outer triangle) on 3, 4, 5, ... vertices are:

1, 1, 3, 12, 55, 273, 1428, 7752, 43263, 246675, ... (sequence A001764 in OEIS),

a sequence that also counts ternary trees and dissections of convex polygons into odd-sided polygons. For instance, there are 12 6-vertex Apollonian networks: three formed by subdividing the outer triangle once and then subdividing two of the resulting triangles, and nine formed by subdividing the outer triangle once, subdividing one of its triangles, and then subdividing one of the resulting smaller triangles.

4.6 History

Birkhoﬀ (1930) is an early paper that uses a dual form of Apollonian networks, the planar maps formed by repeatedly placing new regions at the vertices of simpler maps, as a class of examples of planar maps with few colorings. Geometric structures closely related to Apollonian networks have been studied in polyhedral combinatorics since at least the early 1960s, when they were used by Grünbaum (1963) to describe graphs that can be realized as the graph of a polytope in only one way, without dimensional or combinatorial ambiguities, and by Moon & Moser (1963) to ﬁnd simplicial polytopes with no long paths. In graph theory, the close connection between planarity and treewidth goes back to Robertson & Seymour (1984), who showed that every minor-closed family of graphs either has bounded 4.7. SEE ALSO 19 treewidth or contains all of the planar graphs. Planar 3-trees, as a class of graphs, were explicitly considered by Hakimi & Schmeichel (1979), Alon & Caro (1984), Patil (1986), and many authors since them. The name “Apollonian network” was given by Andrade et al. (2005) to the networks they studied in which the level of subdivision of triangles is uniform across the network; these networks correspond geometrically to a type of stacked polyhedron called a Kleetope.[22] Other authors applied the same name more broadly to planar 3-trees in their work generalizing the model of Andrade et al. to random Apollonian networks.[18] The triangulations generated in this way have also been named “stacked triangulations”[23] or “stack-triangulations”.[24]

4.7 See also

• Barycentric subdivision, a diﬀerent method of subdividing triangles into smaller triangles

• Loop subdivision surface, yet another method of subdividing triangles into smaller triangles

4.8 Notes

[1] This graph is named after the work of Goldner & Harary (1975); however, it appears earlier in the literature, for instance in Grünbaum (1967), p. 357.

[2] The equivalence of planar 3-trees and chordal maximal planar graphs was stated without proof by Patil (1986). For a proof, see Markenzon, Justel & Paciornik (2006). For a more general characterization of chordal planar graphs, and an eﬃcient recognition algorithm for these graphs, see Kumar & Madhavan (1989). The observation that every chordal polyhedral graph is maximal planar was stated explicitly by Gerlach (2004).

[3] Fowler (1998).

[4] The fact that Apollonian networks minimize the number of colorings with more than four of colors was shown in a dual form for colorings of maps by Birkhoﬀ (1930).

[5] Felsner & Zickfeld (2008); Bernardi & Bonichon (2009).

[6] The two forbidden minors for planar graphs are given by Wagner’s theorem. For the forbidden minors for partial 3-trees (which include also the nonplanar Wagner graph) see Arnborg, Proskurowski & Corniel (1986) and Bodlaender (1998). For direct proofs that the octahedral graph and the pentagonal-prism graph are the only two planar forbidden minors, see Dai & Sato (1990) and El-Mallah & Colbourn (1990).

[7] Politof (1983) introduced the Δ-Y reducible planar graphs and characterized them in terms of forbidden homeomorphic subgraphs. The duality between the Δ-Y and Y-Δ reducible graphs, the forbidden minor characterizations of both classes, and the connection to planar partial 3-trees are all from El-Mallah & Colbourn (1990).

[8] For the characterization in terms of the maximum number of triangles in a planar graph, see Hakimi & Schmeichel (1979). Alon & Caro (1984) quote this result and provide the characterizations in terms of the isomorphism classes of blocks and numbers of blocks. The bound on the total number of cliques follows easily from the bounds on triangles and K4 subgraphs, and is also stated explicitly by Wood (2007), who provides an Apollonian network as an example showing that this bound is tight. For generalizations of these bounds to nonplanar surfaces, see Dujmović et al. (2009).

[9] Andrade et al. (2005).

[10] Thurston (1978–1981).

[11] See, e.g., Below, De Loera & Richter-Gebert (2000).

[12] Demaine & Schulz (2011).

[13] Biedl & Ruiz Velázquez (2010).

[14] For subdividing a triangle with rational side lengths so that the smaller triangles also have rational side lengths, see Almering (1963). For progress on the general problem of ﬁnding planar drawings with rational edge lengths, see Geelen, Guo & McKinnon (2008).

[15] For the drawings with integer coordinates, see Mondal et al. (2010), and for the drawings on a given vertex set, see Nishat, Mondal & Rahman (2011). 20 CHAPTER 4. APOLLONIAN NETWORK

[16] Jiménez & Kiwi (2010).

[17] Tsourakakis (2011)

[18] Zhou et al. (2004); Zhou, Yan & Wang (2005).

[19] Frieze & Tsourakakis (2011)

[20] Albenque & Marckert (2008);Zhang et al. (2008).

[21] See Nishizeki (1980) for a 1-tough non-Hamiltonian example, Böhme, Harant & Tkáč (1999) for the proof that Apollonian networks with greater toughness are Hamiltonian, and Gerlach (2004) for an extension of this result to a wider class of planar graphs.

[22] Grünbaum (1963); Grünbaum (1967).

[23] Alon & Caro (1984); Zickfeld & Ziegler (2006); Badent et al. (2007); Felsner & Zickfeld (2008).

[24] Albenque & Marckert (2008); Bernardi & Bonichon (2009); Jiménez & Kiwi (2010).

4.9 References

• Albenque, Marie; Marckert, Jean-François (2008), “Some families of increasing planar maps”, Electronic Journal of Probability 13: 1624–1671, arXiv:0712.0593, doi:10.1214/ejp.v13-563, MR 2438817

• Almering, J. H. J. (1963), “Rational quadrilaterals”, Indagationes Mathematicae 25: 192–199, MR 0147447.

• Alon, N.; Caro, Y. (1984), “On the number of subgraphs of prescribed type of planar graphs with a given number of vertices”, in Rosenfeld, M.; Zaks, J., Convexity and Graph Theory: proceedings of the Conference on Convexity and Graph Theory, Israel, March 1981, Annals of Discrete Mathematics 20, North-Holland Mathematical Studies 87, Elsevier, pp. 25–36, ISBN 978-0-444-86571-7, MR 0791009.

• Andrade, José S., Jr.; Herrmann, Hans J.; Andrade, Roberto F. S.; da Silva, Luciano R. (2005), “Apollonian Networks: Simultaneously Scale-Free, Small World, Euclidean, Space Filling, and with Matching Graphs”, Physics Review Letters 94: 018702, arXiv:cond-mat/0406295.

• Arnborg, S.; Proskurowski, A.; Corniel, D. (1986), Forbidden Minors Characterization of Partial 3-trees, Tech- nical Report CIS-TR-86-07, Dept. of Computer and Information Science, University of Oregon. As cited by El-Mallah & Colbourn (1990).

• Badent, Melanie; Binucci, Carla; Di Giacomo, Emilio; Didimo, Walter; Felsner, Stefan; Giordano, Francesco; Kratochvíl, Jan; Palladino, Pietro; Patrignani, Maurizio; Trotta, Francesco (2007), “Homothetic triangle con- tact representations of planar graphs”, Canadian Conference on Computational Geometry (PDF).

• Below, Alexander; De Loera, Jesús A.; Richter-Gebert, Jürgen (2000), The Complexity of Finding Small Tri- angulations of Convex 3-Polytopes, arXiv:math/0012177.

• Bernardi, Olivier; Bonichon, Nicolas (2009), “Intervals in Catalan lattices and realizers of triangulations”, Journal of Combinatorial Theory, Series A 116 (1): 55–75, doi:10.1016/j.jcta.2008.05.005, MR 2469248.

• Biedl, Therese; Ruiz Velázquez, Lesvia Elena (2010), “Drawing planar 3-trees with given face-areas”, Graph Drawing, 17th International Symposium, GD 2009, Chicago, IL, USA, September 22-25, 2009, Revised Papers, Lecture Notes in Computer Science 5849, Springer-Verlag, pp. 316–322, doi:10.1007/978-3-642-11805- 0_30.

• Birkhoﬀ, George D. (1930), “On the number of ways of colouring a map”, Proceedings of the Edinburgh Mathematical Society, (2) 2: 83–91, doi:10.1017/S0013091500007598.

• Bodlaender, Hans L. (1998), “A partial k-arboretum of graphs with bounded treewidth”, Theoretical Computer Science 209 (1–2): 1–45, doi:10.1016/S0304-3975(97)00228-4, MR 1647486.

• Böhme, Thomas; Harant, Jochen; Tkáč, Michal (1999), “More than one tough chordal planar graphs are Hamil- tonian”, Journal of Graph Theory 32 (4): 405–410, doi:10.1002/(SICI)1097-0118(199912)32:4<405::AID- JGT8>3.3.CO;2-Q, MR 1722793. 4.9. REFERENCES 21

• Butler, S.; Graham, Ron (2010), “Iterated triangle partitions”, in Katona, G.; Schrijver, A.; Szonyi, T., Fete of Combinatorics and Computer Science (PDF), Bolyai Society Mathematical Studies 29, Heidelberg: Springer- Verlag, pp. 23–42.

• Dai, Wayne Wei-Ming; Sato, Masao (1990), “Minimal forbidden minor characterization of planar partial 3- trees and application to circuit layout”, IEEE International Symposium on Circuits and Systems 4, pp. 2677– 2681, doi:10.1109/ISCAS.1990.112560

• Demaine, Erik; Schulz, André (2011), “Embedding stacked polytopes on a polynomial-size grid”, Proc. ACM- SIAM Symposium on Discrete Algorithms (PDF), pp. 1177–1187.

• Dujmović, Vida; Fijavž, Gašper; Joret, Gwenaël; Wood, David R. (2009), The maximum number of cliques in a graph embedded in a surface, arXiv:0906.4142.

• El-Mallah, Ehab S.; Colbourn, Charles J. (1990), “On two dual classes of planar graphs”, Discrete Mathematics 80 (1): 21–40, doi:10.1016/0012-365X(90)90293-Q, MR 1045921.

• Elcock, E. W.; Gargantini, I.; Walsh, T. R. (1987), “Triangular decomposition”, Image and Vision Computing 5 (3): 225–231, doi:10.1016/0262-8856(87)90053-9.

• Felsner, Stefan; Zickfeld, Florian (2008), “On the number of planar orientations with prescribed degrees” (PDF), Electronic Journal of Combinatorics 15 (1): R77, arXiv:math/0701771, MR 2411454.

• Frieze, Alan M.; Tsourakakis, Charalampos E. (2011), High Degree Vertices, Eigenvalues and Diameter of Random Apollonian Networks, arXiv:1104.5259.

• Fowler, Thomas (1998), Unique Coloring of Planar Graphs (PDF), Ph.D. thesis, Georgia Institute of Technol- ogy Mathematics Department.

• Geelen, Jim; Guo, Anjie; McKinnon, David (2008), “Straight line embeddings of cubic planar graphs with integer edge lengths”, Journal of Graph Theory 58 (3): 270–274, doi:10.1002/jgt.20304, MR 2419522.

• Gerlach, T. (2004), “Toughness and Hamiltonicity of a class of planar graphs”, Discrete Mathematics 286 (1-2): 61–65, doi:10.1016/j.disc.2003.11.046, MR 2084280.

• Goldner, A.; Harary, F. (1975), “Note on a smallest nonhamiltonian maximal planar graph”, Bull. Malaysian Math. Soc. 6 (1): 41–42. See also the same journal 6(2):33 (1975) and 8:104-106 (1977). Reference from listing of Harary’s publications.

• Grünbaum, Branko (1963), “Unambiguous polyhedral graphs”, Israel Journal of Mathematics 1 (4): 235–238, doi:10.1007/BF02759726, MR 0185506.

• Grünbaum, Branko (1967), Convex Polytopes, Wiley Interscience.

• Hakimi, S. L.; Schmeichel, E. F. (1979), “On the number of cycles of length k in a maximal planar graph”, Journal of Graph Theory 3 (1): 69–86, doi:10.1002/jgt.3190030108, MR 519175.

• Jiménez, Andrea; Kiwi, Marcos (2010), Counting perfect matchings of cubic graphs in the geometric dual, arXiv:1010.5918.

• Kumar, P. Sreenivasa; Madhavan, C. E. Veni (1989), “A new class of separators and planarity of chordal graphs”, Foundations of Software Technology and Theoretical Computer Science, Ninth Conference, Bangalore, India December 19–21, 1989, Proceedings, Lecture Notes in Computer Science 405, Springer-Verlag, pp. 30–43, doi:10.1007/3-540-52048-1_30, MR 1048636.

• Markenzon, L.; Justel, C. M.; Paciornik, N. (2006), “Subclasses of k-trees: Characterization and recognition”, Discrete Applied Mathematics 154 (5): 818–825, doi:10.1016/j.dam.2005.05.021, MR 2207565.

• Mondal, Debajyoti; Nishat, Rahnuma Islam; Rahman, Md. Saidur; Alam, Muhammad Jawaherul (2010), “Minimum-area drawings of plane 3-trees”, Canadian Conference on Computational Geometry (PDF).

• Moon, J. W.; Moser, L. (1963), “Simple paths on polyhedra”, Paciﬁc Journal of Mathematics 13: 629–631, doi:10.2140/pjm.1963.13.629, MR 0154276. 22 CHAPTER 4. APOLLONIAN NETWORK

• Nishat, Rahnuma Islam; Mondal, Debajyoti; Rahman, Md. Saidur (2011), “Point-set embeddings of plane 3-trees”, Graph Drawing, 18th International Symposium, GD 2010, Konstanz, Germany, September 21-24, 2010, Revised Selected Papers, Lecture Notes in Computer Science 6502, Springer-Verlag, pp. 317–328, doi:10.1007/978-3-642-18469-7_29. • Nishizeki, Takao (1980), “A 1-tough non-Hamiltonian maximal planar graph”, Discrete Mathematics 30 (3): 305–307, doi:10.1016/0012-365X(80)90240-X, MR 0573648. • Patil, H. P. (1986), “On the structure of k-trees”, Journal of Combinatorics, Information and System Sciences 11 (2-4): 57–64, MR 966069. • Petersen, Julius (1891), “Die Theorie der regulären graphs”, Acta Mathematica 15: 193–220, doi:10.1007/BF02392606.

• Plummer, Michael D. (1992), “Extending matchings in planar graphs IV”, Discrete Mathematics 109 (1–3): 207–219, doi:10.1016/0012-365X(92)90292-N, MR 1192384.

• Politof, T. (1983), A characterization and eﬃcient reliability computation of Δ-Y reducible networks, Ph.D. thesis, University of California, Berkeley. As cited by El-Mallah & Colbourn (1990). • Robertson, Neil; Seymour, P. D. (1984), “Graph minors. III. Planar tree-width”, Journal of Combinatorial Theory, Series B 36 (1): 49–64, doi:10.1016/0095-8956(84)90013-3, MR 0742386. • Takeo, Fujio (1960), “On triangulated graphs. I”, Bull. Fukuoka Gakugei Univ. III 10: 9–21, MR 0131372. An error regarding Hamiltonicity was pointed out by MathSciNet reviewer W. T. Tutte. • Thurston, William (1978–1981), The geometry and topology of 3-manifolds, Princeton lecture notes.

• Tsourakakis, Charalampos E. (2011), The Degree Sequence of Random Apollonian Networks. • Wood, David R. (2007), “On the maximum number of cliques in a graph”, Graphs and Combinatorics 23 (3): 337–352, arXiv:math/0602191, doi:10.1007/s00373-007-0738-8, MR 2320588. • Zhang, Zhongzhi; Chen, Lichao; Zhou, Shuigeng; Fang, Lujun; Guan, Jihong; Zou, Tao (2008), “Analytical solution of average path length for Apollonian networks”, Physical Review E 77: 017102, arXiv:0706.3491, doi:10.1103/PhysRevE.77.017102.

• Zhou, Tao; Yan, Gang; Wang, Bing-Hong (2005), “Maximal planar networks with large clustering coef- ﬁcient and power-law degree distribution”, Physical Review E 71 (4): 046141, arXiv:cond-mat/0412448, doi:10.1103/PhysRevE.71.046141. • Zhou, Tao; Yan, Gang; Zhou, Pei-Ling; Fu, Zhong-Qian; Wang, Bing-Hong (2004), Random Apollonian networks, arXiv:cond-mat/0409414v2. • Zickfeld, Florian; Ziegler, Günter M. (2006), “Integer realizations of stacked polytopes”, Workshop on Geo- metric and Topological Combinatorics (PDF).

4.10 External links

• Weisstein, Eric W., “Apollonian Network”, MathWorld. • Matlab Simulation Code 4.10. EXTERNAL LINKS 23

Construction of an Apollonian network from a circle packing 24 CHAPTER 4. APOLLONIAN NETWORK

The triakis tetrahedron, a polyhedral realization of an 8-vertex Apollonian network Chapter 5

Asymmetric graph

The eight 6-vertex asymmetric graphs

In graph theory, a branch of mathematics, an undirected graph is called an asymmetric graph if it has no nontrivial symmetries. Formally, an automorphism of a graph is a permutation p of its vertices with the property that any two vertices u and v are adjacent if and only if p(u) and p(v) are adjacent. The identity mapping of a graph onto itself is always an automorphism, and is called the trivial automorphism of the graph. An asymmetric graph is a graph for which there are no other automorphisms.

5.1 Examples

The smallest asymmetric non-trivial graphs have 6 vertices.[1] The smallest asymmetric regular graphs have ten vertices; there exist ten-vertex asymmetric graphs that are 4-regular and 5-regular.[2][3] One of the two smallest asymmetric cubic graphs is the twelve-vertex Frucht graph discovered in 1939.[4] According to a strengthened version of Frucht’s theorem, there are inﬁnitely many asymmetric cubic graphs.

25 26 CHAPTER 5. ASYMMETRIC GRAPH

The Frucht graph, the smallest asymmetric cubic graph.

5.2 Properties

The class of asymmetric graphs is closed under complements: a graph G is asymmetric if and only if its complement is.[1] Any n-vertex asymmetric graph can be made symmetric by adding and removing a total of at most n/2 + o(n) edges.[1]

5.3 Random graphs

The proportion of graphs on n vertices with nontrivial automorphism tends to zero as n grows, which is informally expressed as "almost all finite graphs are asymmetric”. In contrast, again informally, “almost all infinite graphs are symmetric.” More specifically, countable infinite random graphs in the Erdős–Rényi model are, with probability 1, isomorphic to the highly symmetric Rado graph.[1]

5.4 Trees

The smallest asymmetric tree has seven vertices: it consists of three paths of lengths 1, 2, and 3, linked at a common endpoint.[5] In contrast to the situation for graphs, almost all trees are symmetric. In particular, if a tree is chosen 5.5. REFERENCES 27 uniformly at random among all trees on n labeled nodes, then with probability tending to 1 as n increases, the tree will contain some two leaves adjacent to the same node and will have symmetries exchanging these two leaves.[1]

5.5 References

[1] Erdős, P.; Rényi, A. (1963), “Asymmetric graphs” (PDF), Acta Mathematica Hungarica 14 (3): 295–315, doi:10.1007/BF01895716.

[2] Baron, G.; Imrich, W. (1969), “Asymmetrische reguläre Graphen”, Acta Mathematica Academiae Scientiarum Hungaricae 20: 135–142, doi:10.1007/BF01894574, MR 0238726.

[3] Gewirtz, Allan; Hill, Anthony; Quintas, Louis V. (1969), “The minimal number of points for regular asymmetric graphs”, Universidad Técnica Federico Santa Maria. Scientia 138: 103–111, MR 0266818.

[4] Frucht, R. (1939), “Herstellung von Graphen mit vorgegebener abstrakter Gruppe.”, Compositio Mathematica (in German) 6: 239–250, ISSN 0010-437X, Zbl 0020.07804.

[5] Quintas, Louis V. (1967), “Extrema concerning asymmetric graphs”, Journal of Combinatorial Theory 3 (1): 57–82, doi:10.1016/S0021-9800(67)80018-8. Chapter 6

Biased graph

In mathematics, a biased graph is a graph with a list of distinguished circles (edge sets of simple cycles), such that if two circles in the list are contained in a theta graph, then so is the third circle of the theta graph. A biased graph is a generalization of the combinatorial essentials of a gain graph and in particular of a signed graph. Formally, a biased graph Ω is a pair (G, B) where B is a linear class of circles; this by deﬁnition is a class of circles that satisﬁes the theta-graph property mentioned above. A subgraph or edge set whose circles are all in B (and which contains no half-edges is called balanced. For instance, a circle belonging to B is balanced and one that does not belong to B is unbalanced. Biased graphs are interesting mostly because of their matroids, but also because of their connection with multiary quasigroups. See below.

6.1 Technical notes

A biased graph may have half-edges (one endpoint) and loose edges (no endpoints). The edges with two endpoints are of two kinds: a link has two distinct endpoints, while a loop has two coinciding endpoints. Linear classes of circles are a special case of linear subclasses of circuits in a matroid.

6.2 Examples

• If every circle belongs to B, and there are no half-edges, Ω is balanced. A balanced biased graph is (for most purposes) essentially the same as an ordinary graph.

• If B is empty, Ω is called contrabalanced. Contrabalanced biased graphs are related to bicircular matroids.

• If B consists of the circles of even length, Ω is called antibalanced and is the biased graph obtained from an all-negative signed graph.

• The linear class B is additive, that is, closed under repeated symmetric diﬀerence (when the result is a circle), if and only if B is the class of positive circles of a signed graph.

• Ω may have underlying graph that is a cycle of length n ≥ 3 with all edges doubled. Call this a biased 2Cn . Such biased graphs in which no digon (circle of length 2) is balanced lead to spikes and swirls (see Matroids, below).

• Some kinds of biased graph are obtained from gain graphs or are generalizations of special kinds of gain graph. The latter include biased expansion graphs, which generalize group expansion graphs.

28 6.3. MINORS 29

6.3 Minors

A minor of a biased graph Ω = (G, B) is the result of any sequence of taking subgraphs and contracting edge sets. For biased graphs, as for graphs, it suﬃces to take a subgraph (which may be the whole graph) and then contract an edge set (which may be the empty set). A subgraph of Ω consists of a subgraph H of the underlying graph G, with balanced circle class consisting of those balanced circles that are in H. The deletion of an edge set S, written Ω − S, is the subgraph with all vertices and all edges except those of S. Contraction of Ω is relatively complicated. To contract one edge e, the procedure depends on the kind of edge e is. If e is a link, contract it in G. A circle C in the contraction G/e is balanced if either C or C ∪ e is a balanced circle of G. If e is a balanced loop or a loose edge, it is simply deleted. If it is an unbalanced loop or a half-edge, it and its vertex v are deleted; each other edge with v as an endpoint loses that endpoint, so a link with v as one endpoint becomes a half-edge at its other endpoint, while a loop or half-edge at v becomes a loose edge. In the contraction Ω/S by an arbitrary edge set S, the edge set is E − S. (We let G = (V, E).) The vertex set is the class of vertex sets of balanced components of the subgraph (V, S) of Ω. That is, if (V, S) has balanced components with vertex sets V1, ..., Vk, then Ω/S has k vertices V1, ..., Vk . An edge e of Ω, not in S, becomes an edge of Ω/S and each endpoint vi of e in Ω that belongs to some Vi becomes the endpoint Vi of e in Ω/S ; thus, an endpoint of e that is not in a balanced component of (V, S) disappears. An edge with all endpoints in unbalanced components of (V, S) becomes a loose edge in the contraction. An edge with only one endpoint in a balanced component of (V, S) becomes a half-edge. An edge with two endpoints that belong to diﬀerent balanced components becomes a link, and an edge with two endpoints that belong to the same balanced component becomes a loop.

6.4 Matroids

There are two kinds of matroid associated with a biased graph, both of which generalize the cycle matroid of a graph (Zaslavsky, 1991).

6.4.1 The frame matroid

The frame matroid (sometimes called bias matroid) of a biased graph, M(Ω), (Zaslavsky, 1989) has for its ground set the edge set E. An edge set is independent if each component contains either no circles or just one circle, which is unbalanced. (In matroid theory a half-edge acts like an unbalanced loop and a loose edge acts like a balanced loop.) M(Ω) is a frame matroid in the abstract sense, meaning that it is a submatroid of a matroid in which, for at least one basis, the set of lines generated by pairs of basis elements covers the whole matroid. Conversely, every abstract frame matroid is the frame matroid of some biased graph. The circuits of the matroid are called frame circuits or bias circuits. There are four kinds. One is a balanced circle. Two other kinds are a pair of unbalanced circles together with a connecting simple path, such that the two circles are either disjoint (then the connecting path has one end in common with each circle and is otherwise disjoint from both) or share just a single common vertex (in this case the connecting path is that single vertex). The fourth kind of circuit is a theta graph in which every circle is unbalanced. The rank of an edge set S is n − b, where n is the number of vertices of G and b is the number of balanced components of S, counting isolated vertices as balanced components. Minors of the frame matroid agree with minors of the biased graph; that is, M(Ω−S) = M(Ω)−S and M(Ω/S) = M(Ω)/S. Frame matroids generalize the Dowling geometries associated with a group (Dowling, 1973). The frame matroid of a biased 2Cn (see Examples, above) which has no balanced digons is called a swirl. It is important in matroid structure theory.

6.4.2 The lift matroid

The extended lift matroid L0(Ω) has for its ground set the set E0, which is the union of E with an extra point e0. The lift matroid L(Ω) is the extended lift matroid restricted to E. The extra point acts exactly like an unbalanced 30 CHAPTER 6. BIASED GRAPH

loop or a half-edge, so we describe only the lift matroid. An edge set is independent if it contains either no circles or just one circle, which is unbalanced. A circuit is a balanced circle, a pair of unbalanced circles that are either disjoint or have just a common vertex, or a theta graph whose circles are all unbalanced. The rank of an edge set S is n − c + ε, where c is the number of components of S, counting isolated vertices, and ε is 0 if S is balanced and 1 if it is not. Minors of the lift and extended lift matroids agree in part with minors of the biased graph. Deletions agree: L(Ω−S) = L(Ω)−S. Contractions agree only for balanced edge sets: M(Ω/S) = M(Ω)/S if S is balanced, but not if it is unbalanced. If S is unbalanced, M(Ω/S) = M(G)/S = M(G/S) where M of a graph denotes the ordinary graphic matroid. The lift matroid of a 2Cn (see Examples, above) which has no balanced digons is called a spike. Spikes are quite important in matroid structure theory.

6.5 Multiary quasigroups

Just as a group expansion of a complete graph Kn encodes the group (see Dowling geometry), its combinatorial analog expanding a simple cycle of length n + 1 encodes an n-ary (multiary) quasigroup. It is possible to prove theorems about multiary quasigroups by means of biased graphs (Zaslavsky, t.a.)

6.6 References

• T. A. Dowling (1973), A class of geometric lattices based on ﬁnite groups. Journal of Combinatorial Theory Series B, Vol. 14, 61–86. • Thomas Zaslavsky (1989), Biased graphs. I. Bias, balance, and gains. Journal of Combinatorial Theory Series B, Vol. 47, 32–52. • Thomas Zaslavsky (1991), Biased graphs. II. The three matroids. Journal of Combinatorial Theory Series B, Vol. 51, 46–72. • Thomas Zaslavsky (1999). A mathematical bibliography of signed and gain graphs and allied areas. Electronic Journal of Combinatorics, Dynamic Surveys in Combinatorics, #DS8. • Thomas Zaslavsky (t.a.), Associativity in multiary quasigroups: The way of biased expansions. Aequationes Mathematicae, to appear. Chapter 7

Biclique-free graph

In graph theory, a branch of mathematics, a t-biclique-free graph is a graph that has no 2t-vertex complete bipartite graph Kt,t as a subgraph. A family of graphs is biclique-free if there exists a number t such that the graphs in the family are all t-biclique-free. The biclique-free graph families form one of the most general types of sparse graph family. They arise in incidence problems in discrete geometry, and have also been used in parameterized complexity.

7.1 Properties

7.1.1 Sparsity

According to the Kővári–Sós–Turán theorem, every n-vertex t-biclique-free graph has O(n2 − 1/t) edges, signiﬁcantly fewer than a dense graph would have.[1] Conversely, if a graph family is deﬁned by forbidden subgraphs or closed under the operation of taking subgraphs, and does not include dense graphs of arbitrarily large size, it must be t- biclique-free for some t, for otherwise it would include large dense complete bipartite graphs. As a lower bound, Erdős, Hajnal & Moon (1964) conjectured that every maximal t-biclique-free bipartite graph (one to which no more edges can be added without creating a t-biclique) has at least (t − 1)(n + m − t + 1) edges, where n and m are the numbers of vertices on each side of its bipartition.[2]

7.1.2 Relation to other types of sparse graph family

A graph with degeneracy d is necessarily (d + 1)-biclique-free. Additionally, a biclique-free family of graphs must be nowhere dense, meaning that for every number k, there exists a graph that is not a k-shallow minor of a graph in the family. In particular, if there exists an n-vertex graph that is not a 1-shallow minor, then the family must be n- biclique-free, because all n-vertex graphs are 1-shallow minors of Kn,n. In this way, the biclique-free graph families unify two of the most general classes of sparse graphs.[3]

7.2 Applications

7.2.1 Discrete geometry

In discrete geometry, many types of incidence graph are necessarily biclique-free. As a simple example, the graph of incidences between a ﬁnite set of points and lines in the Euclidean plane necessarily has no K₂,₂ subgraph.[4]

7.2.2 Parameterized complexity

Biclique-free graphs have been used in parameterized complexity to develop algorithms that are efficient for sparse graphs with suitably small input parameter values. In particular, finding a dominating set of size k, on t-biclique-free graphs, is fixed-parameter tractable when parameterized by k + t, even though there is strong evidence that this is not

31 32 CHAPTER 7. BICLIQUE-FREE GRAPH possible using k alone as a parameter. Similar results are true for many variations of the dominating set problem.[3] It is also possible to test whether one dominating set of size at most k can be converted to another one by a chain of vertex insertions and deletions, preserving the dominating property, with the same parameterization.[5]

7.3 References

[1] Kővári, T.; T. Sós, V.; Turán, P. (1954), “On a problem of K. Zarankiewicz” (PDF), Colloquium Math. 3: 50–57, MR 0065617. This work concerns the number of edges in biclique-free bipartite graphs, but a standard application of the probabilistic method transfers the same bound to arbitrary graphs.

[2] Erdős, P.; Hajnal, A.; Moon, J. W. (1964), “A problem in graph theory” (PDF), The American Mathematical Monthly 71: 1107–1110, doi:10.2307/2311408, MR 0170339.

[3] Telle, Jan Arne; Villanger, Yngve (2012), “FPT algorithms for domination in biclique-free graphs”, in Epstein, Leah; Ferragina, Paolo, Algorithms – ESA 2012: 20th Annual European Symposium, Ljubljana, Slovenia, September 10–12, 2012, Proceedings, Lecture Notes in Computer Science 7501, Springer, pp. 802–812, doi:10.1007/978-3-642-33090-2_69.

[4] Kaplan, Haim; Matoušek, Jiří; Sharir, Micha (2012), “Simple proofs of classical theorems in discrete geometry via the Guth–Katz polynomial partitioning technique”, Discrete and Computational Geometry 48 (3): 499–517, arXiv:1102.5391, doi:10.1007/s00454-012-9443-3, MR 2957631. See in particular Lemma 3.1 and the remarks following the lemma.

[5] Lokshtanov, Daniel; Mouawad, Amer E.; Panolan, Fahad; Ramanujan, M. S.; Saurabh, Saket (2015), “Reconﬁguration on sparse graphs”, in Dehne, Frank; Sack, Jörg-Rüdiger; Stege, Ulrike, Algorithms and Data Structures: 14th International Symposium, WADS 2015, Victoria, BC, Canada, August 5-7, 2015, Proceedings (PDF), Lecture Notes in Computer Science 9214, Springer, pp. 506–517, arXiv:1502.04803, doi:10.1007/978-3-319-21840-3_42. Chapter 8

Biconnected graph

In graph theory, a biconnected graph is a connected and “nonseparable” graph, meaning that if any vertex were to be removed, the graph will remain connected. Therefore a biconnected graph has no articulation vertices. The property of being 2-connected is equivalent to biconnectivity, with the caveat that the complete graph of two vertices is sometimes regarded as biconnected but not 2-connected. This property is especially useful in maintaining a graph with a two-fold redundancy, to prevent disconnection upon the removal of a single edge (or connection). The use of biconnected graphs is very important in the ﬁeld of networking (see Network ﬂow), because of this property of redundancy.

8.1 Deﬁnition

A biconnected undirected graph is a connected graph that is not broken into disconnected pieces by deleting any single vertex (and its incident edges). A biconnected directed graph is one such that for any two vertices v and w there are two directed paths from v to w which have no vertices in common other than v and w.

8.2 Examples

•

8.3 See also

• Biconnected component

8.4 References

• Eric W. Weisstein. “Biconnected Graph.” From MathWorld--A Wolfram Web Resource. http://mathworld. wolfram.com/BiconnectedGraph.html

• Paul E. Black, “biconnected graph”, in Dictionary of Algorithms and Data Structures [online], Paul E. Black, ed., U.S. National Institute of Standards and Technology. 17 December 2004. (accessed TODAY) Available from: http://www.nist.gov/dads/HTML/biconnectedGraph.html

33 34 CHAPTER 8. BICONNECTED GRAPH

8.5 External links

• The tree of the biconnected components Java implementation in the jBPT library (see BCTree class). Chapter 9

Bipartite graph

U V

Example of a bipartite graph without cycles

In the mathematical ﬁeld of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint sets U and V (that is, U and V are each independent sets) such that every edge connects a vertex in U to one in V . Vertex set U and V are usually called the parts of the graph. Equivalently, a bipartite graph is a

35 36 CHAPTER 9. BIPARTITE GRAPH

A complete bipartite graph with m = 5 and n = 3 graph that does not contain any odd-length cycles.[1][2] The two sets U and V may be thought of as a coloring of the graph with two colors: if one colors all nodes in U blue, and all nodes in V green, each edge has endpoints of diﬀering colors, as is required in the graph coloring problem.[3][4] In contrast, such a coloring is impossible in the case of a non-bipartite graph, such as a triangle: after one node is colored blue and another green, the third vertex of the triangle is connected to vertices of both colors, preventing it from being assigned either color. One often writes G = (U, V, E) to denote a bipartite graph whose partition has the parts U and V , with E denoting the edges of the graph. If a bipartite graph is not connected, it may have more than one bipartition;[5] in this case, the (U, V, E) notation is helpful in specifying one particular bipartition that may be of importance in an application. If |U| = |V | , that is, if the two subsets have equal cardinality, then G is called a balanced bipartite graph.[3] If all vertices on the same side of the bipartition have the same degree, then G is called biregular.

9.1 Examples

When modelling relations between two different classes of objects, bipartite graphs very often arise naturally. For instance, a graph of football players and clubs, with an edge between a player and a club if the player has played for that club, is a natural example of an affiliation network, a type of bipartite graph used in social network analysis.[6] Another example where bipartite graphs appear naturally is in the (NP-complete) railway optimization problem, in which the input is a schedule of trains and their stops, and the goal is to find a set of train stations as small as possible such that every train visits at least one of the chosen stations. This problem can be modeled as a dominating set problem in a bipartite graph that has a vertex for each train and each station and an edge for each pair of a station and a train that stops at that station.[7] A third example is in the academic field of numismatics. Ancient coins are made using two positive impressions of the design (the obverse and reverse). The charts numismatists produce to represent the production of coins are bipartite graphs. [8] More abstract examples include the following:

• Every tree is bipartite.[4]

• Cycle graphs with an even number of vertices are bipartite.[4]

• Every planar graph whose faces all have even length is bipartite.[9] Special cases of this are grid graphs and squaregraphs, in which every inner face consists of 4 edges and every inner vertex has four or more neighbors.[10] 9.2. PROPERTIES 37

• The complete bipartite graph on m and n vertices, denoted by Kn,m is the bipartite graph G = (U, V, E), where U and V are disjoint sets of size m and n, respectively, and E connects every vertex in U with all vertices in V. It follows that Km,n has mn edges.[11] Closely related to the complete bipartite graphs are the crown graphs, formed from complete bipartite graphs by removing the edges of a perfect matching.[12]

• Hypercube graphs, partial cubes, and median graphs are bipartite. In these graphs, the vertices may be labeled by bitvectors, in such a way that two vertices are adjacent if and only if the corresponding bitvectors diﬀer in a single position. A bipartition may be formed by separating the vertices whose bitvectors have an even number of ones from the vertices with an odd number of ones. Trees and squaregraphs form examples of median graphs, and every median graph is a partial cube.[13]

9.2 Properties

9.2.1 Characterization

Bipartite graphs may be characterized in several diﬀerent ways:

• A graph is bipartite if and only if it does not contain an odd cycle.[14]

• A graph is bipartite if and only if it is 2-colorable, (i.e. its chromatic number is less than or equal to 2).[3]

• The spectrum of a graph is symmetric if and only if it’s a bipartite graph.[15]

9.2.2 König’s theorem and perfect graphs

In bipartite graphs, the size of minimum vertex cover is equal to the size of the maximum matching; this is König’s theorem.[16][17] An alternative and equivalent form of this theorem is that the size of the maximum independent set plus the size of the maximum matching is equal to the number of vertices. In any graph without isolated vertices the size of the minimum edge cover plus the size of a maximum matching equals the number of vertices.[18] Combining this equality with König’s theorem leads to the facts that, in bipartite graphs, the size of the minimum edge cover is equal to the size of the maximum independent set, and the size of the minimum edge cover plus the size of the minimum vertex cover is equal to the number of vertices. Another class of related results concerns perfect graphs: every bipartite graph, the complement of every bipartite graph, the line graph of every bipartite graph, and the complement of the line graph of every bipartite graph, are all perfect. Perfection of bipartite graphs is easy to see (their chromatic number is two and their maximum clique size is also two) but perfection of the complements of bipartite graphs is less trivial, and is another restatement of König’s theorem. This was one of the results that motivated the initial deﬁnition of perfect graphs.[19] Perfection of the complements of line graphs of perfect graphs is yet another restatement of König’s theorem, and perfection of the line graphs themselves is a restatement of an earlier theorem of König, that every bipartite graph has an edge coloring using a number of colors equal to its maximum degree. According to the strong perfect graph theorem, the perfect graphs have a forbidden graph characterization resembling that of bipartite graphs: a graph is bipartite if and only if it has no odd cycle as a subgraph, and a graph is perfect if and only if it has no odd cycle or its complement as an induced subgraph. The bipartite graphs, line graphs of bipartite graphs, and their complements form four out of the ﬁve basic classes of perfect graphs used in the proof of the strong perfect graph theorem.[20]

9.2.3 Degree

For a vertex, the number of adjacent vertices is called the degree of the vertex and is denoted deg(v) . The degree sum formula for a bipartite graph states that

∑ ∑ deg(v) = deg(u) = |E| . v∈V u∈U 38 CHAPTER 9. BIPARTITE GRAPH

The degree sequence of a bipartite graph is the pair of lists each containing the degrees of the two parts U and V . For example, the complete bipartite graph K₃,₅ has degree sequence (5, 5, 5), (3, 3, 3, 3, 3) . Isomorphic bipartite graphs have the same degree sequence. However, the degree sequence does not, in general, uniquely identify a bipartite graph; in some cases, non-isomorphic bipartite graphs may have the same degree sequence. The bipartite realization problem is the problem of ﬁnding a simple bipartite graph with the degree sequence being two given lists of natural numbers. (Trailing zeros may be ignored since they are trivially realized by adding an appropriate number of isolated vertices to the digraph.)

9.2.4 Relation to hypergraphs and directed graphs

The biadjacency matrix of a bipartite graph (U, V, E) is a (0,1)-matrix of size |U|×|V | that has a one for each pair of adjacent vertices and a zero for nonadjacent vertices.[21] Biadjacency matrices may be used to describe equivalences between bipartite graphs, hypergraphs, and directed graphs. A hypergraph is a combinatorial structure that, like an undirected graph, has vertices and edges, but in which the edges may be arbitrary sets of vertices rather than having to have exactly two endpoints. A bipartite graph (U, V, E) may be used to model a hypergraph in which U is the set of vertices of the hypergraph, V is the set of hyperedges, and E contains an edge from a hypergraph vertex v to a hypergraph edge e exactly when v is one of the endpoints of v . Under this correspondence, the biadjacency matrices of bipartite graphs are exactly the incidence matrices of the corresponding hypergraphs. As a special case of this correspondence between bipartite graphs and hypergraphs, any multigraph (a graph in which there may be two or more edges between the same two vertices) may be interpreted as a hypergraph in which some hyperedges have equal sets of endpoints, and represented by a bipartite graph that does not have multiple adjacencies and in which the vertices on one side of the bipartition all have degree two.[22] A similar reinterpretation of adjacency matrices may be used to show a one-to-one correspondence between directed graphs (on a given number of labeled vertices, allowing self-loops) and balanced bipartite graphs, with the same number of vertices on both sides of the bipartition. For, the adjacency matrix of a directed graph with n vertices can be any (0, 1) -matrix of size n × n , which can then be reinterpreted as the adjacency matrix of a bipartite graph with n vertices on each side of its bipartition.[23] In this construction, the bipartite graph is the bipartite double cover of the directed graph.

9.3 Algorithms

9.3.1 Testing bipartiteness

It is possible to test whether a graph is bipartite, and to return either a two-coloring (if it is bipartite) or an odd cycle (if it is not) in linear time, using depth-first search. The main idea is to assign to each vertex the color that differs from the color of its parent in the depth-first search tree, assigning colors in a preorder traversal of the depth-first-search tree. This will necessarily provide a two-coloring of the spanning tree consisting of the edges connecting vertices to their parents, but it may not properly color some of the non-tree edges. In a depth-first search tree, one of the two endpoints of every non-tree edge is an ancestor of the other endpoint, and when the depth first search discovers an edge of this type it should check that these two vertices have different colors. If they do not, then the path in the tree from ancestor to descendant, together with the miscolored edge, form an odd cycle, which is returned from the algorithm together with the result that the graph is not bipartite. However, if the algorithm terminates without detecting an odd cycle of this type, then every edge must be properly colored, and the algorithm returns the coloring together with the result that the graph is bipartite.[24] Alternatively, a similar procedure may be used with breadth-first search in place of depth-first search. Again, each node is given the opposite color to its parent in the search tree, in breadth-first order. If, when a vertex is colored, there exists an edge connecting it to a previously-colored vertex with the same color, then this edge together with the paths in the breadth-first search tree connecting its two endpoints to their lowest common ancestor forms an odd cycle. If the algorithm terminates without finding an odd cycle in this way, then it must have found a proper coloring, and can safely conclude that the graph is bipartite.[25] For the intersection graphs of n line segments or other simple shapes in the Euclidean plane, it is possible to test whether the graph is bipartite and return either a two-coloring or an odd cycle in time O(n log n) , even though the graph itself may have as many as Ω(n2) edges.[26] 9.3. ALGORITHMS 39

9.3.2 Odd cycle transversal

A graph with an odd cycle transversal of size 2: removing the two blue bottom vertices leaves a bipartite graph.

Odd cycle transversal is an NP-complete algorithmic problem that asks, given a graph G = (V,E) and a number k, whether there exists a set of k vertices whose removal from G would cause the resulting graph to be bipartite.[27] The problem is fixed-parameter tractable, meaning that there is an algorithm whose running time can be bounded by a polynomial function of the size of the graph multiplied by a larger function of k.[28] More specifically, the time for this algorithm is O(3k mn), although this was not stated in that paper.[29] The result by Reed et al. was obtained using a completely new method, which later was called iterative compression and turned out to be an extremely useful algorithmic tool, especially in the field of fixed-parameter tractability. This tool is now considered one of the fundamental tools for parameterized algorithmics. The name odd cycle transversal comes from the fact that a graph is bipartite if and only if it has no odd cycles. Hence, to delete vertices from a graph in order to obtain a bipartite graph, one needs to “hit all odd cycle”, or find a so-called odd cycle transversal set. In the illustration, one can observe that every odd cycle in the graph contains the blue (the bottommost) vertices, hence removing those vertices kills all odd cycles and leaves a bipartite graph. The edge bipartization problem is the algorithmic problem of deleting as few edges as possible to make a graph bipartite and is also an important problem in graph modification algorithmics. This problem is also fixed-parameter tractable, and can be solved in time O(2k m2),[30] where k is the number of edges to delete and m is the number of edges in the input graph. 40 CHAPTER 9. BIPARTITE GRAPH

9.3.3 Matching

A matching in a graph is a subset of its edges, no two of which share an endpoint. Polynomial time algorithms are known for many algorithmic problems on matchings, including maximum matching (finding a matching that uses as many edges as possible), maximum weight matching, and stable marriage.[31] In many cases, matching problems are simpler to solve on bipartite graphs than on non-bipartite graphs,[32] and many matching algorithms such as the Hopcroft–Karp algorithm for maximum cardinality matching[33] work correctly only on bipartite inputs. As a simple example, suppose that a set P of people are all seeking jobs from among a set of J jobs, with not all people suitable for all jobs. This situation can be modeled as a bipartite graph (P, J, E) where an edge connects each job-seeker with each suitable job.[34] A perfect matching describes a way of simultaneously satisfying all job-seekers and filling all jobs; Hall’s marriage theorem provides a characterization of the bipartite graphs which allow perfect matchings. The National Resident Matching Program applies graph matching methods to solve this problem for U.S. medical student job-seekers and hospital residency jobs.[35] The Dulmage–Mendelsohn decomposition is a structural decomposition of bipartite graphs that is useful in finding maximum matchings.[36]

9.4 Additional applications

Bipartite graphs are extensively used in modern coding theory, especially to decode codewords received from the channel. Factor graphs and Tanner graphs are examples of this. A Tanner graph is a bipartite graph in which the vertices on one side of the bipartition represent digits of a codeword, and the vertices on the other side represent combinations of digits that are expected to sum to zero in a codeword without errors.[37] A factor graph is a closely related belief network used for probabilistic decoding of LDPC and turbo codes.[38] In computer science, a Petri net is a mathematical modeling tool used in analysis and simulations of concurrent systems. A system is modeled as a bipartite directed graph with two sets of nodes: A set of “place” nodes that contain resources, and a set of “event” nodes which generate and/or consume resources. There are additional constraints on the nodes and edges that constrain the behavior of the system. Petri nets utilize the properties of bipartite directed graphs and other properties to allow mathematical proofs of the behavior of systems while also allowing easy implementation of simulations of the system.[39] In projective geometry, Levi graphs are a form of bipartite graph used to model the incidences between points and lines in a conﬁguration. Corresponding to the geometric property of points and lines that every two lines meet in at most one point and every two points be connected with a single line, Levi graphs necessarily do not contain any cycles of length four, so their girth must be six or more.[40]

9.5 See also

• Bipartite dimension, the minimum number of complete bipartite graphs whose union is the given graph

• Bipartite double cover, a way of transforming any graph into a bipartite graph by doubling its vertices

• Bipartite matroid, a class of matroids that includes the graphic matroids of bipartite graphs

• Bipartite network projection, a weighting technique for compressing information about bipartite networks

• Convex bipartite graph, a bipartite graph whose vertices can be ordered so that the vertex neighborhoods are contiguous

• Multipartite graph, a generalization of bipartite graphs to more than two subsets of vertices

• Quasi-bipartite graph, a type of Steiner tree problem instance in which the terminals form an independent set, allowing approximation algorithms that generalize those for bipartite graphs

• Split graph, a graph in which the vertices can be partitioned into two subsets, one of which is independent and the other of which is a clique

• Zarankiewicz problem on the maximum number of edges in a bipartite graph with forbidden subgraphs 9.6. REFERENCES 41

9.6 References

[1] Diestel, Reinard (2005). Graph Theory, Grad. Texts in Math. Springer. ISBN 978-3-642-14278-9.

[2] Asratian, Armen S.; Denley, Tristan M. J.; Häggkvist, Roland (1998), Bipartite Graphs and their Applications, Cambridge Tracts in Mathematics 131, Cambridge University Press, ISBN 9780521593458.

[3] Asratian, Denley & Häggkvist (1998), p. 7.

[4] Scheinerman, Edward R. (2012), Mathematics: A Discrete Introduction (3rd ed.), Cengage Learning, p. 363, ISBN 9780840049421.

[5] Chartrand, Gary; Zhang, Ping (2008), Chromatic Graph Theory, Discrete Mathematics And Its Applications 53, CRC Press, p. 223, ISBN 9781584888000.

[6] Wasserman, Stanley; Faust, Katherine (1994), Social Network Analysis: Methods and Applications, Structural Analysis in the Social Sciences 8, Cambridge University Press, pp. 299–302, ISBN 9780521387071.

[7] Niedermeier, Rolf (2006). Invitation to Fixed Parameter Algorithms (Oxford Lecture Series in Mathematics and Its Appli- cations). Oxford. pp. 20–21. ISBN 978-0-19-856607-6.

[8] Bracey, Robert (2012). “On the Graphical Interpreation of Herod’s Coinage in Judaea and Rome in Coins”. pp. 65–84.

[9] Soifer, Alexander (2008), The Mathematical Coloring Book, Springer-Verlag, pp. 136–137, ISBN 978-0-387-74640-1. This result has sometimes been called the “two color theorem"; Soifer credits it to a famous 1879 paper of Alfred Kempe containing a false proof of the four color theorem.

[10] Bandelt, H.-J.; Chepoi, V.; Eppstein, D. (2010), “Combinatorics and geometry of ﬁnite and inﬁnite squaregraphs”, SIAM Journal on Discrete Mathematics 24 (4): 1399–1440, arXiv:0905.4537, doi:10.1137/090760301.

[11] Asratian, Denley & Häggkvist (1998), p. 11.

[12] Archdeacon, D.; Debowsky, M.; Dinitz, J.; Gavlas, H. (2004), “Cycle systems in the complete bipartite graph minus a one-factor”, Discrete Mathematics 284 (1–3): 37–43, doi:10.1016/j.disc.2003.11.021.

[13] Ovchinnikov, Sergei (2011), Graphs and Cubes, Universitext, Springer. See especially Chapter 5, “Partial Cubes”, pp. 127–181.

[14] Asratian, Denley & Häggkvist (1998), Theorem 2.1.3, p. 8. Asratian et al. attribute this characterization to a 1916 paper by Dénes Kőnig. For inﬁnite graphs, this result requires the axiom of choice.

[15] Biggs, Norman (1994), Algebraic Graph Theory, Cambridge Mathematical Library (2nd ed.), Cambridge University Press, p. 53, ISBN 9780521458979.

[16] Kőnig, Dénes (1931). “Gráfok és mátrixok”. Matematikai és Fizikai Lapok 38: 116–119..

[17] Gross, Jonathan L.; Yellen, Jay (2005), Graph Theory and Its Applications, Discrete Mathematics And Its Applications (2nd ed.), CRC Press, p. 568, ISBN 9781584885054.

[18] Chartrand, Gary; Zhang, Ping (2012), A First Course in Graph Theory, Courier Dover Publications, pp. 189–190, ISBN 9780486483689.

[19] Béla Bollobás (1998), Modern Graph Theory, Graduate Texts in Mathematics 184, Springer, p. 165, ISBN 9780387984889.

[20] Chudnovsky, Maria; Robertson, Neil; Seymour, Paul; Thomas, Robin (2006), “The strong perfect graph theorem”, Annals of Mathematics 164 (1): 51–229, doi:10.4007/annals.2006.164.51.

[21] Asratian, Denley & Häggkvist (1998), p. 17.

[22] A. A. Sapozhenko (2001), “Hypergraph”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1- 55608-010-4

[23] Brualdi, Richard A.; Harary, Frank; Miller, Zevi (1980), “Bigraphs versus digraphs via matrices”, Journal of Graph The- ory 4 (1): 51–73, doi:10.1002/jgt.3190040107, MR 558453. Brualdi et al. credit the idea for this equivalence to Dul- mage, A. L.; Mendelsohn, N. S. (1958), “Coverings of bipartite graphs”, Canadian Journal of Mathematics 10: 517–534, doi:10.4153/CJM-1958-052-0, MR 0097069.

[24] Sedgewick, Robert (2004), Algorithms in Java, Part 5: Graph Algorithms (3rd ed.), Addison Wesley, pp. 109–111.

[25] Kleinberg, Jon; Tardos, Éva (2006), Algorithm Design, Addison Wesley, pp. 94–97. 42 CHAPTER 9. BIPARTITE GRAPH

[26] Eppstein, David (2009), “Testing bipartiteness of geometric intersection graphs”, ACM Transactions on Algorithms 5 (2): Art. 15, arXiv:cs.CG/0307023, doi:10.1145/1497290.1497291, MR 2561751.

[27] Yannakakis, Mihalis (1978), “Node-and edge-deletion NP-complete problems”, Proceedings of the 10th ACM Symposium on Theory of Computing (STOC '78), pp. 253–264, doi:10.1145/800133.804355

[28] Reed, Bruce; Smith, Kaleigh; Vetta, Adrian (2004), “Finding odd cycle transversals”, Operations Research Letters 32 (4): 299–301, doi:10.1016/j.orl.2003.10.009, MR 2057781.

[29] Hüﬀner, Falk (2005), “Algorithm engineering for optimal graph bipartization”, Experimental and Eﬃcient Algorithms: 240–252, doi:10.1007/11427186_22.

[30] Guo, Jiong; Gramm, Jens; Hüﬀner, Falk; Niedermeier, Rolf; Wernicke, Sebastian (2006), “Compression-based ﬁxed- parameter algorithms for feedback vertex set and edge bipartization”, JCSS: 1386–1396, doi:10.1016/j.jcss.2006.02.001

[31] Ahuja, Ravindra K.; Magnanti, Thomas L.; Orlin, James B. (1993), “12. Assignments and Matchings”, Network Flows: Theory, Algorithms, and Applications, Prentice Hall, pp. 461–509.

[32] Ahuja, Magnanti & Orlin (1993), p. 463: “Nonbipartite matching problems are more diﬃcult to solve because they do not reduce to standard network ﬂow problems.”

[33] Hopcroft, John E.; Karp, Richard M. (1973), “An n5/2 algorithm for maximum matchings in bipartite graphs”, SIAM Journal on Computing 2 (4): 225–231, doi:10.1137/0202019.

[34] Ahuja, Magnanti & Orlin (1993), Application 12.1 Bipartite Personnel Assignment, pp. 463–464.

[35] Robinson, Sara (April 2003), “Are Medical Students Meeting Their (Best Possible) Match?" (PDF), SIAM News (3): 36.

[36] Dulmage & Mendelsohn (1958).

[37] Moon, Todd K. (2005), Error Correction Coding: Mathematical Methods and Algorithms, John Wiley & Sons, p. 638, ISBN 9780471648000.

[38] Moon (2005), p. 686.

[39] Cassandras, Christos G.; Lafortune, Stephane (2007), Introduction to Discrete Event Systems (2nd ed.), Springer, p. 224, ISBN 9780387333328.

[40] Grünbaum, Branko (2009), Conﬁgurations of Points and Lines, Graduate Studies in Mathematics 103, American Mathe- matical Society, p. 28, ISBN 9780821843086.

9.7 External links

• Hazewinkel, Michiel, ed. (2001), “Graph, bipartite”, Encyclopedia of Mathematics, Springer, ISBN 978-1- 55608-010-4 • Information System on Graph Classes and their Inclusions: bipartite graph

• Weisstein, Eric W., “Bipartite Graph”, MathWorld. Chapter 10 k-Variegated graph

In graph theory, a bivariegated graph is a graph whose vertex set can be partitioned into two equal parts such that each vertex is adjacent to exactly one vertex from the other set not containing it.[1][2][3] In a bivarigated graph G with 2n vertices, there exists a set of n independent edges such that no odd number of them lie on a cycle of G.

10.1 Examples

The Petersen graph, shown below, is a bivariegated graph: if one partitions it into an outer pentagon and an inner ﬁve-point star, each vertex on one side of the partition has exactly one neighbor on the other side of the partition. More generally, the same is true for any generalized Petersen graph formed by connecting an outer polygon and an inner star with the same number of points; for instance, this applies to the Möbius–Kantor graph and the Desargues graph. Any hypercube graph, such as the four-dimensional hypercube shown below, is also bivariegated. However, the graph shown below is not bivariegated. Whatever you choose the three independent edges, one of them is an edge of a cycle.

10.2 Bivariegated trees

A tree T with 2n vertices, is bivariegated if and only if the independence number of T is n, or, equivalently, if and only if it has a perfect matching.[1]

10.3 Generalizations

The k-varigated graph, k ≥ 3, can be deﬁned similarly. A graph is said to be k-varigated if its vertex set can be partitioned into k equal parts such that each vertex is adjacent to exactly one vertex from every other part not containing it.[2]

10.4 Notes

• Characterizing the degree sequences of bivariegated graphs has been an unsolved problem in graph theory.

[1] Bednarek & Sanders (1973).

[2] Bhat-Nayak, Choudum & Naik (1978).

[3] Riddle (1978).

43 44 CHAPTER 10. K-VARIEGATED GRAPH

10.5 References

• Bednarek, A. R.; Sanders, E. L. (1973), “A characterization of bivariegated trees”, Discrete Mathematics 5: 1–14, doi:10.1016/0012-365X(73)90022-8. • Bhat-Nayak, Vasanti N.; Choudum, S. A.; Naik, Ranjan N. (1978), “Characterization of 2-variegated graphs and of 3-variegated graphs”, Discrete Mathematics 23: 17–22, doi:10.1016/0012-365X(78)90182-6. • Bhat-Nayak, Vasanti N.; Kocay, W. L.; Naik, Ranjan N. (1980), “Forcibly 2-variegated degree sequences”, Utilitas Math. 18: 83–89. • Bhat-Nayak Vasanti N., Ranjan N. Naik, Further results on 2-variegated graphs, Utilitas Math. 12 (1977) 317–325. • Javdekar, Medha (1980), “Characterization of forcibly k-variegated degree sequences, k ≥ 3”, Discrete Math- ematics 29 (1): 33–38, doi:10.1016/0012-365X(90)90284-O. • Javdekar, Medha (1980), “Characterization of k-variegated graphs, k ≥ 3”, Discrete Mathematics 32 (3): 263– 270, doi:10.1016/0012-365X(80)90264-2 • Riddle, Fay A. (1978), Bivariegated Graphs and Their Isomorphisms, Ph.D. dissertation, University of Florida. 10.5. REFERENCES 45 46 CHAPTER 10. K-VARIEGATED GRAPH

6 5 4 1

3 2 Chapter 11

Block graph

Not to be confused with block diagram or bar chart. In graph theory, a branch of combinatorial mathematics, a block graph or clique tree[1] is a type of undirected

A block graph graph in which every biconnected component (block) is a clique. Block graphs are sometimes erroneously called Husimi trees (after Kôdi Husimi),[2] but that name more properly refers to cactus graphs, graphs in which every nontrivial biconnected component is a cycle.[3] Block graphs may be characterized as the intersection graphs of the blocks of arbitrary undirected graphs.[4]

11.1 Characterization

Block graphs are exactly the graphs for which, for every four vertices u, v, x, and y, the largest two of the three distances d(u,v) + d(x,y), d(u,x) + d(v,y), and d(u,y) + d(v,x) are always equal.[2][5] They also have a forbidden graph characterization as the graphs that do not have the diamond graph or a cycle of four or more vertices as an induced subgraph; that is, they are the diamond-free chordal graphs.[5] They are also the

47 48 CHAPTER 11. BLOCK GRAPH

Ptolemaic graphs (chordal distance-hereditary graphs) in which every two nodes at distance two from each other are connected by a unique shortest path,[2] and the chordal graphs in which every two maximal cliques have at most one vertex in common.[2] A graph G is a block graph if and only if the intersection of every two connected subsets of vertices of G is empty or connected. Therefore, the connected subsets of vertices in a connected block graph form a convex geometry, a property that is not true of any graphs that are not block graphs.[6] Because of this property, in a connected block graph, every set of vertices has a unique minimal connected superset, its closure in the convex geometry. The connected block graphs are exactly the graphs in which there is a unique induced path connecting every pair of vertices.[1]

11.2 Related graph classes

Block graphs are chordal and distance-hereditary. The distance-hereditary graphs are the graphs in which every two induced paths between the same two vertices have the same length, a weakening of the characterization of block graphs as having at most one induced path between every two vertices. Because both the chordal graphs and the distance-hereditary graphs are subclasses of the perfect graphs, block graphs are perfect. Every tree is a block graph. Another class of examples of block graphs is provided by the windmill graphs. Every block graph has boxicity at most two.[7] Block graphs are examples of pseudo-median graphs: for every three vertices, either there exists a unique vertex that belongs to shortest paths between all three vertices, or there exists a unique triangle whose edges lie on these three shortest paths.[7] The line graphs of trees are exactly the block graphs in which every cut vertex is incident to at most two blocks, or equivalently the claw-free block graphs. Line graphs of trees have been used to ﬁnd graphs with a given number of edges and vertices in which the largest induced subgraph that is a tree is as small as possible.[8]

11.3 Block graphs of undirected graphs

If G is any undirected graph, the block graph of G, denoted B(G), is the intersection graph of the blocks of G: B(G) has a vertex for every biconnected component of G, and two vertices of B(G) are adjacent if the corresponding two blocks meet at an articulation vertex. If K1 denotes the graph with one vertex, then B(K1) is deﬁned to be the empty graph. B(G) is necessarily a block graph: it has one biconnected component for each articulation vertex of G, and each biconnected component formed in this way must be a clique. Conversely, every block graph is the graph B(G) for some graph G.[4] If G is a tree, then B(G) coincides with the line graph of G. The graph B(B(G)) has one vertex for each articulation vertex of G; two vertices are adjacent in B(B(G)) if they belong to the same block in G.[4]

11.4 References

[1] Vušković, Kristina (2010), “Even-hole-free graphs: A survey”, Applicable Analysis and Discrete Mathematics 4 (2): 219– 240, doi:10.2298/AADM100812027V.

[2] Howorka, Edward (1979), “On metric properties of certain clique graphs”, Journal of Combinatorial Theory, Series B 27 (1): 67–74, doi:10.1016/0095-8956(79)90069-8.

[3] See, e.g., MR 0659742, a 1983 review by Robert E. Jamison of another paper referring to block graphs as Husimi trees; Jamison attributes the mistake to an error in a book by Behzad and Chartrand.

[4] Harary, Frank (1963), “A characterization of block-graphs”, Canadian Mathematical Bulletin 6 (1): 1–6, doi:10.4153/cmb- 1963-001-x.

[5] Bandelt, Hans-Jürgen; Mulder, Henry Martyn (1986), “Distance-hereditary graphs”, Journal of Combinatorial Theory, Series B 41 (2): 182–208, doi:10.1016/0095-8956(86)90043-2.

[6] Edelman, Paul H.; Jamison, Robert E. (1985), “The theory of convex geometries”, Geometriae Dedicata 19 (3): 247–270, doi:10.1007/BF00149365. 11.4. REFERENCES 49

[7] Block graphs, Information System on Graph Class Inclusions.

[8] Erdős, Paul; Saks, Michael; Sós, Vera T. (1986), “Maximum induced trees in graphs”, Journal of Combinatorial Theory, Series B 41 (1): 61–79, doi:10.1016/0095-8956(86)90028-6. Chapter 12

Bound graph

In graph theory, a bound graph expresses which pairs of elements of some partially ordered set have an upper bound. Rigorously, any graph G is a bound graph if there exists a partial order ≤ on the vertices of G with the property that for any vertices u and v of G, uv is an edge of G if and only if u ≠ v and there is a vertex w such that u ≤ w and v ≤ w. Bound graphs are sometimes referred to as upper bound graphs, but the analogously deﬁned lower bound graphs comprise exactly the same class—any lower bound for ≤ is easily seen to be an upper bound for the dual partial order ≥.

12.1 References

• McMorris, F.R.; Zaslavsky, T. (1982). “Bound graphs of a partially ordered set”. Journal of Combinatorics, Information & System Sciences 7: 134–138.

• Lundgren, J.R.; Maybee, J.S. (1983). “A characterization of upper bound graphs”. Congressus Numerantium 40: 189–193.

• Bergstrand, D.J.; Jones, K.F. (1988). “On upper bound graphs of partially ordered sets”. Congressus Numer- antium 66: 185–193.

• Tanenbaum, P.J. (2000). “Bound graph polysemy” (PDF). Electronic Journal of Combinatorics 7: #R43.

50 Chapter 13

Cactus graph

In graph theory, a cactus (sometimes called a cactus tree) is a connected graph in which any two simple cycles have at most one vertex in common. Equivalently, every edge in such a graph belongs to at most one simple cycle. Equivalently, every block (maximal subgraph without a cut-vertex) is an edge or a cycle.

13.1 Properties

Cacti are outerplanar graphs. Every pseudotree is a cactus. A graph is a cactus if and only if every block is either a simple cycle or a single edge. The family of graphs in which each component is a cactus is downwardly closed under graph minor operations. This graph family may be characterized by a single forbidden minor, the four-vertex diamond graph formed by removing [1] an edge from the complete graph K4.

13.2 Algorithms and applications

Some facility location problems which are NP-hard for general graphs, as well as some other graph problems, may be solved in polynomial time for cacti.[2][3] Since cacti are special cases of outerplanar graphs, a number of combinatorial optimization problems on graphs may be solved for them in polynomial time.[4] Cacti represent electrical circuits that have useful properties. An early application of cacti was associated with the representation of op-amps.[5][6][7] Cacti have also recently been used in comparative genomics as a way of representing the relationship between diﬀerent genomes or parts of genomes.[8] If a cactus is connected, and each of its vertices belongs to at most two blocks, then it is called a Christmas cactus. Every polyhedral graph has a Christmas cactus subgraph that includes all of its vertices, a fact that plays an essential role in a proof by Leighton & Moitra (2010) that every polyhedral graph has a greedy embedding in the Euclidean plane, an assignment of coordinates to the vertices for which greedy forwarding succeeds in routing messages between all pairs of vertices.[9]

13.3 History

Cacti were ﬁrst studied under the name of Husimi trees, bestowed on them by Frank Harary and George Eugene Uhlenbeck in honor of previous work on these graphs by Kôdi Husimi.[10][11] The same Harary–Uhlenbeck paper reserves the name “cactus” for graphs of this type in which every cycle is a triangle, but now allowing cycles of all lengths is standard. Meanwhile, the name Husimi tree commonly came to refer to graphs in which every block is a complete graph

51 52 CHAPTER 13. CACTUS GRAPH

A cactus graph 13.4. REFERENCES 53

(equivalently, the intersection graphs of the blocks in some other graph). This usage had little to do with the work of Husimi, and the more pertinent term block graph is now used for this family; however, because of this ambiguity this phrase has become less frequently used to refer to cactus graphs.[12]

13.4 References

[1] El-Mallah, Ehab; Colbourn, Charles J. (1988), “The complexity of some edge deletion problems”, IEEE Transactions on Circuits and Systems 35 (3): 354–362, doi:10.1109/31.1748

[2] Ben-Moshe, Boaz; Bhattacharya, Binay; Shi, Qiaosheng (2005), “Eﬃcient algorithms for the weighted 2-center problem in a cactus graph”, Algorithms and Computation, 16th Int. Symp., ISAAC 2005, Lecture Notes in Computer Science 3827, Springer-Verlag, pp. 693–703, doi:10.1007/11602613_70

[3] Zmazek, Blaz; Zerovnik, Janez (2005), “Estimating the traﬃc on weighted cactus networks in linear time”, Ninth Interna- tional Conference on Information Visualisation (IV'05), pp. 536–541, doi:10.1109/IV.2005.48, ISBN 0-7695-2397-8

[4] Korneyenko, N. M. (1994), “Combinatorial algorithms on a class of graphs”, Discrete Applied Mathematics 54 (2–3): 215– 217, doi:10.1016/0166-218X(94)90022-1. Translated from Notices of the BSSR Academy of Sciences, Ser. Phys.-Math. Sci., (1984) no. 3, pp. 109-111 (Russian)

[5] Nishi, Tetsuo; Chua, Leon O. (1986), “Topological proof of the Nielsen-Willson theorem”, IEEE Transactions on Circuits and Systems 33 (4): 398–405, doi:10.1109/TCS.1986.1085935

[6] Nishi, Tetsuo; Chua, Leon O. (1986), “Uniqueness of solution for nonlinear resistive circuits containing CCCS’s or VCVS’s whose controlling coeﬃcients are ﬁnite”, IEEE Transactions on Circuits and Systems 33 (4): 381–397, doi:10.1109/TCS.1986.1085934

[7] Nishi, Tetsuo (1991), “On the number of solutions of a class of nonlinear resistive circuit”, Proceedings of the IEEE Inter- national Symposium on Circuits and Systems, Singapore, pp. 766–769

[8] Paten, Benedict; Diekhans, Mark; Earl, Dent; St. John, John; Ma, Jian; Suh, Bernard; Haussler, David (2010), “Research in Computational Molecular Biology”, Lecture Notes in Computer Science, Lecture Notes in Computer Science 6044: 410– 425, doi:10.1007/978-3-642-12683-3_27, ISBN 978-3-642-12682-6 |chapter= ignored (help)

[9] Leighton, Tom; Moitra, Ankur (2010), “Some Results on Greedy Embeddings in Metric Spaces” (PDF), Discrete & Com- putational Geometry 44 (3): 686–705, doi:10.1007/s00454-009-9227-6.

[10] Harary, Frank; Uhlenbeck, George E. (1953), “On the number of Husimi trees, I”, Proceedings of the National Academy of Sciences 39 (4): 315–322, doi:10.1073/pnas.39.4.315, MR 0053893

[11] Husimi, Kodi (1950), “Note on Mayers’ theory of cluster integrals”, Journal of Chemical Physics 18 (5): 682–684, doi:10.1063/1.1747725, MR 0038903

[12] See, e.g., MR 0659742, a 1983 review by Robert E. Jamison of a paper using the other deﬁnition, which attributes the ambiguity to an error in a book by Behzad and Chartrand.

13.5 External links

• Application of Cactus Graphs in Analysis and Design of Electronic Circuits Chapter 14

Cage (graph theory)

The Tutte (3,8)-cage.

In the mathematical area of graph theory, a cage is a regular graph that has as few vertices as possible for its girth. Formally, an (r,g)-graph is deﬁned to be a graph in which each vertex has exactly r neighbors, and in which the shortest cycle has length exactly g. It is known that an (r,g)-graph exists for any combination of r ≥ 2 and g ≥ 3. An (r,g)-cage

54 14.1. KNOWN CAGES 55

is an (r,g)-graph with the fewest possible number of vertices, among all (r,g)-graphs. If a Moore graph exists with degree r and girth g, it must be a cage. Moreover, the bounds on the sizes of Moore graphs generalize to cages: any cage with odd girth g must have at least

(g∑−3)/2 1 + r (r − 1)i i=0 vertices, and any cage with even girth g must have at least

(g∑−2)/2 2 (r − 1)i i=0 vertices. Any (r,g)-graph with exactly this many vertices is by deﬁnition a Moore graph and therefore automatically a cage. There may exist multiple cages for a given combination of r and g. For instance there are three nonisomorphic (3,10)- cages, each with 70 vertices : the Balaban 10-cage, the Harries graph and the Harries–Wong graph. But there is only one (3,11)-cage : the Balaban 11-cage (with 112 vertices).

14.1 Known cages

A degree-one graph has no cycle, and a connected degree-two graph has girth equal to its number of vertices, so cages are only of interest for r ≥ 3. The (r,3)-cage is a complete graph Kr₊₁ on r+1 vertices, and the (r,4)-cage is a complete bipartite graph Kr,r on 2r vertices. Other notable cages include the Moore graphs:

• (3,5)-cage: the Petersen graph, 10 vertices

• (3,6)-cage: the Heawood graph, 14 vertices

• (3,8)-cage: the Tutte–Coxeter graph, 30 vertices

• (3,10)-cage: the Balaban 10-cage, 70 vertices

• (3,11)-cage: the Balaban 11-cage, 112 vertices

• (4,5)-cage: the Robertson graph, 19 vertices

• (7,5)-cage: The Hoﬀman–Singleton graph, 50 vertices.

• When r − 1 is a prime power, the (r,6) cages are the incidence graphs of projective planes.

• When r − 1 is a prime power, the (r,8) and (r,12) cages are generalized polygons.

The numbers of vertices in the known (r,g) cages, for values of r > 2 and g > 2, other than projective planes and generalized polygons, are:

14.2 Asymptotics

For large values of g, the Moore bound implies that the number n of vertices must grow at least singly exponentially as a function of g. Equivalently, g can be at most proportional to the logarithm of n. More precisely,

≤ g 2 logr−1 n + O(1). 56 CHAPTER 14. CAGE (GRAPH THEORY)

It is believed that this bound is tight or close to tight (Bollobás & Szemerédi 2002). The best known lower bounds on g are also logarithmic, but with a smaller constant factor (implying that n grows singly exponentially but at a higher rate than the Moore bound). Speciﬁcally, the Ramanujan graphs (Lubotzky, Phillips & Sarnak 1988) satisfy the bound

4 g ≥ log n + O(1). 3 r−1 It is unlikely that these graphs are themselves cages, but their existence gives an upper bound to the number of vertices needed in a cage.

14.3 References

• Biggs, Norman (1993), Algebraic Graph Theory (2nd ed.), Cambridge Mathematical Library, pp. 180–190, ISBN 0-521-45897-8.

• Bollobás, Béla; Szemerédi, Endre (2002), “Girth of sparse graphs”, Journal of Graph Theory 39 (3): 194–200, doi:10.1002/jgt.10023, MR 1883596.

• Exoo, G; Jajcay, R (2008), “Dynamic Cage Survey”, Electronic Journal of Combinatorics (Dynamic Survey) DS16.

• Erdős, Paul; Rényi, Alfréd; Sós, Vera T. (1966), “On a problem of graph theory” (PDF), Studia Sci. Math. Hungar. 1: 215–235.

• Hartsﬁeld, Nora; Ringel, Gerhard (1990), Pearls in Graph Theory: A Comprehensive Introduction, Academic Press, pp. 77–81, ISBN 0-12-328552-6.

• Holton, D. A.; Sheehan, J. (1993), The Petersen Graph, Cambridge University Press, pp. 183–213, ISBN 0-521-43594-3. • Lubotzky, A.; Phillips, R.; Sarnak, P. (1988), “Ramanujan graphs”, Combinatorica 8 (3): 261–277, doi:10.1007/BF02126799, MR 963118. • Tutte, W. T. (1947), “A family of cubical graphs”, Proc. Cambridge Philos. Soc. 43 (4): 459–474, doi:10.1017/S0305004100023720.

14.4 External links

• Brouwer, Andries E. Cages • Royle, Gordon. Cubic Cages and Higher valency cages

• Weisstein, Eric W., “Cage Graph”, MathWorld. Chapter 15

Cayley graph

e a

The Cayley graph of the free group on two generators a and b

In mathematics, a Cayley graph, also known as a Cayley colour graph, Cayley diagram, group diagram, or colour group[1] is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley’s theorem (named after Arthur Cayley) and uses a specified, usually finite, set of generators for the group. It is a

57 58 CHAPTER 15. CAYLEY GRAPH

central tool in combinatorial and geometric group theory.

15.1 Deﬁnition

Suppose that G is a group and S is a generating set. The Cayley graph Γ = Γ(G, S) is a colored directed graph constructed as follows:[2]

• Each element g of G is assigned a vertex: the vertex set V (Γ) of Γ is identiﬁed with G.

• Each generator s of S is assigned a color cs . • For any g ∈ G, s ∈ S, the vertices corresponding to the elements g and gs are joined by a directed edge of colour cs. Thus the edge set E(Γ) consists of pairs of the form (g, gs), with s ∈ S providing the color.

In geometric group theory, the set S is usually assumed to be ﬁnite, symmetric (i.e. S = S−1 ) and not containing the identity element of the group. In this case, the uncolored Cayley graph is an ordinary graph: its edges are not oriented and it does not contain loops (single-element cycles).

15.2 Examples

• Suppose that G = Z is the inﬁnite cyclic group and the set S consists of the standard generator 1 and its inverse (−1 in the additive notation) then the Cayley graph is an inﬁnite path.

• Similarly, if G = Zn is the finite cyclic group of order n and the set S consists of two elements, the standard generator of G and its inverse, then the Cayley graph is the cycle Cn . • The Cayley graph of the direct product of groups (with the cartesian product of generating sets as a generating set) is the cartesian product of the corresponding Cayley graphs.[3] Thus the Cayley graph of the abelian group Z2 with the set of generators consisting of four elements (1, 0), (0, 1) is the infinite grid on the plane R2 , while for the direct product Zn × Zm with similar generators the Cayley graph is the n × m finite grid on a torus.

• A Cayley graph of the dihedral group D4 on two generators a and b is depicted to the left. Red arrows represent left-multiplication by element a. Since element b is self-inverse, the blue lines which represent left- multiplication by element b are undirected. Therefore the graph is mixed: it has eight vertices, eight arrows, and four edges. The Cayley table of the group D4 can be derived from the group presentation

⟨a, b|a4 = b2 = e, ab = ba3⟩.

A different Cayley graph of Dih4 is shown on the right. b is still the horizontal reflection and represented by blue lines; c is a diagonal reflection and represented by green lines. As both reflections are self-inverse the Cayley graph on the right is completely undirected. This graph corresponds to the presentation

⟨b, c|b2 = c2 = e, bcbc = cbcb⟩.

• The Cayley graph of the free group on two generators a, b corresponding to the set S = {a, b, a−1, b−1} is depicted at the top of the article, and e represents the identity element. Travelling along an edge to the right represents right multiplication by a, while travelling along an edge upward corresponds to the multiplication by b. Since the free group has no relations, the Cayley graph has no cycles. This Cayley graph is a key ingredient in the proof of the Banach–Tarski paradox. 15.3. CHARACTERIZATION 59

Cayley graph of the dihedral group Dih4 on two generators a and b     1 x z  •   ∈ Z A Cayley graph of the discrete Heisenberg group  0 1 y , x, y, z  0 0 1

is depicted to the right. The generators used in the picture are the three matrices X, Y, Z given by the three permutations of 1, 0, 0 for the entries x, y, z. They satisfy the relations Z = XYX−1Y −1,XZ = ZX,YZ = ZY , which can also be understood from the picture. This is a non-commutative inﬁnite group, and despite being a three-dimensional space, the Cayley graph has four-dimensional volume growth.

15.3 Characterization

The group G acts on itself by left multiplication (see Cayley’s theorem). This may be viewed as the action of G on its Cayley graph. Explicitly, an element h ∈ G maps a vertex g ∈ V (Γ) to the vertex hg ∈ V (Γ) . The set of edges within the Cayley graph is preserved by this action: the edge (g, gs) is transformed into the edge (hg, hgs) . The left multiplication action of any group on itself is simply transitive, in particular, the Cayley graph is vertex transitive. This leads to the following characterization of Cayley graphs:

Sabidussi Theorem: A graph Γ is a Cayley graph of a group G if and only if it admits a simply transitive 60 CHAPTER 15. CAYLEY GRAPH

On two generators of Dih4, which are both self-inverse

action of G by graph automorphisms (i.e. preserving the set of edges).[4]

To recover the group G and the generating set S from the Cayley graph Γ = Γ(G, S) , select a vertex v1 ∈ V (Γ) and label it by the identity element of the group. Then label each vertex v of Γ by the unique element of G that transforms v1 into v. The set S of generators of G that yields Γ as the Cayley graph is the set of labels of the vertices adjacent to the selected vertex. The generating set is finite (this is a common assumption for Cayley graphs) if and only if the graph is locally finite (i.e. each vertex is adjacent to finitely many edges).

15.4 Elementary properties

• If a member s of the generating set is its own inverse, s = s−1 , then it is typically represented by an undirected edge. • The Cayley graph Γ(G, S) depends in an essential way on the choice of the set S of generators. For example, if the generating set S has k elements then each vertex of the Cayley graph has k incoming and k outgoing directed edges. In the case of a symmetric generating set S with r elements, the Cayley graph is a regular directed graph of degree r. • Cycles (or closed walks) in the Cayley graph indicate relations between the elements of S. In the more elabo- rate construction of the Cayley complex of a group, closed paths corresponding to relations are “ﬁlled in” by polygons. This means that the problem of constructing the Cayley graph of a given presentation P is equivalent to solving the Word Problem for P .[1] • If f : G′ → G is a surjective group homomorphism and the images of the elements of the generating set S′ for G′ are distinct, then it induces a covering of graphs

f¯ : Γ(G′,S′) → Γ(G, S), where S = f(S′).

In particular, if a group G has k generators, all of order diﬀerent from 2, and the set S consists of these generators together with their inverses, then the Cayley graph Γ(G, S) is covered by the inﬁnite regular tree of degree 2k corresponding to the free group on the same set of generators. 15.5. SCHREIER COSET GRAPH 61

Part of a Cayley graph of the Heisenberg group. (The coloring is only for visual aid.)

• A graph Γ(G, S) can be constructed even if the set S does not generate the group G. However, it is disconnected and is not considered to be a Cayley graph. In this case, each connected component of the graph represents a coset of the subgroup generated by S .

• For any ﬁnite Cayley graph, considered as undirected, the vertex connectivity is at least equal to 2/3 of the degree of the graph. If the generating set is minimal (removal of any element and, if present, its inverse from the generating set leaves a set which is not generating), the vertex connectivity is equal to the degree. The edge connectivity is in all cases equal to the degree.[5]

15.5 Schreier coset graph

Main article: Schreier coset graph

If one, instead, takes the vertices to be right cosets of a ﬁxed subgroup H , one obtains a related construction, the Schreier coset graph, which is at the basis of coset enumeration or the Todd–Coxeter process.

15.6 Connection to group theory

Knowledge about the structure of the group can be obtained by studying the adjacency matrix of the graph and in particular applying the theorems of spectral graph theory. 62 CHAPTER 15. CAYLEY GRAPH

15.6.1 Geometric group theory

For infinite groups, the coarse geometry of the Cayley graph is fundamental to geometric group theory. For a finitely generated group, this is independent of choice of finite set of generators, hence an intrinsic property of the group. This is only interesting for infinite groups: every finite group is coarsely equivalent to a point (or the trivial group), since one can choose as finite set of generators the entire group. Formally, for a given choice of generators, one has the word metric (the natural distance on the Cayley graph), which determines a metric space. The coarse equivalence class of this space is an invariant of the group.

15.7 History

The Cayley Graph was ﬁrst considered for ﬁnite groups by Arthur Cayley in 1878.[2] Max Dehn in his unpublished lectures on group theory from 1909–10 reintroduced Cayley graphs under the name Gruppenbild (group diagram), which led to the geometric group theory of today. His most important application was the solution of the word problem for the fundamental group of surfaces with genus ≥ 2, which is equivalent to the topological problem of deciding which closed curves on the surface contract to a point.[6]

15.8 Bethe lattice

Main article: Bethe lattice

The Bethe lattice or Cayley tree, is the Cayley graph of the free group on n generators. A presentation of a group G by n generators corresponds to a surjective map from the free group on n generators to the group G, and at the level of Cayley graphs to a map from the Cayley tree to the Cayley graph. This can also be interpreted (in algebraic topology) as the universal cover of the Cayley graph, which is not in general simply connected.

15.9 See also

• Vertex-transitive graph • Generating set of a group • Lovász conjecture • Cube-connected cycles • Algebraic graph theory • Cycle graph (algebra)

15.10 Notes

[1] Magnus, Wilhelm; Karrass, Abraham; Solitar, Donald (2004) [1966]. Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations. Courier. ISBN 978-0-486-43830-6.

[2] Cayley, Arthur (1878). “Desiderata and suggestions: No. 2. The Theory of groups: graphical representation”. Amer. J. Math. 1 (2): 174–6. doi:10.2307/2369306. JSTOR 2369306. In his Collected Mathematical Papers 10: 403–405.

[3] Theron, Daniel Peter (1988), An extension of the concept of graphically regular representations, Ph.D. thesis, University of Wisconsin, Madison, p. 46, MR 2636729.

[4] Sabidussi, Gert (October 1958). “On a Class of Fixed-Point-Free Graphs”. Proceedings of the American Mathematical Society 9 (5): 800–4. doi:10.1090/s0002-9939-1958-0097068-7. JSTOR 2033090.

[5] See Theorem 3.7 of Babai, László (1995). “Chapter 27: Automorphism groups, isomorphism, reconstruction” (PDF). In Graham, R. L.; Grötschel, M.; Lovász, L. Handbook of Combinatorics. Amsterdam: Elsevier. pp. 1447–1540. 15.11. EXTERNAL LINKS 63

[6] Dehn, Max (2012) [1987]. Papers on Group Theory and Topology. Springer-Verlag. ISBN 1461291070. Translated from the German and with introductions and an appendix by John Stillwell, and with an appendix by Otto Schreier.

15.11 External links

• Cayley diagrams • Weisstein, Eric W., “Cayley graph”, MathWorld. Chapter 16

Chordal graph

A cycle (black) with two chords (green). As for this part, the graph is chordal. However, removing one green edge would result in a non-chordal graph. Indeed, the other green edge with three black edges would form a cycle of length four with no chords.

In the mathematical area of graph theory, a chordal graph is one in which all cycles of four or more vertices have a chord, which is an edge that is not part of the cycle but connects two vertices of the cycle. Equivalently, every induced cycle in the graph should have at most three vertices. The chordal graphs may also be characterized as the graphs that have perfect elimination orderings, as the graphs in which each minimal separator is a clique, and as the intersection graphs of subtrees of a tree. They are sometimes also called rigid circuit graphs[1] or triangulated graphs.[2] Chordal graphs are a subset of the perfect graphs. They may be recognized in polynomial time, and several problems that are hard on other classes of graphs such as graph coloring may be solved in polynomial time when the input is chordal. The treewidth of an arbitrary graph may be characterized by the size of the cliques in the chordal graphs that contain it.

64 16.1. PERFECT ELIMINATION AND EFFICIENT RECOGNITION 65

16.1 Perfect elimination and eﬃcient recognition

A perfect elimination ordering in a graph is an ordering of the vertices of the graph such that, for each vertex v, v and the neighbors of v that occur after v in the order form a clique. A graph is chordal if and only if it has a perfect elimination ordering.[3] Rose, Lueker & Tarjan (1976) (see also Habib et al. 2000) show that a perfect elimination ordering of a chordal graph may be found efficiently using an algorithm known as lexicographic breadth-first search. This algorithm maintains a partition of the vertices of the graph into a sequence of sets; initially this sequence consists of a single set with all vertices. The algorithm repeatedly chooses a vertex v from the earliest set in the sequence that contains previously unchosen vertices, and splits each set S of the sequence into two smaller subsets, the first consisting of the neighbors of v in S and the second consisting of the non-neighbors. When this splitting process has been performed for all vertices, the sequence of sets has one vertex per set, in the reverse of a perfect elimination ordering. Since both this lexicographic breadth first search process and the process of testing whether an ordering is a perfect elimination ordering can be performed in linear time, it is possible to recognize chordal graphs in linear time. The graph sandwich problem on chordal graphs is NP-complete[4] whereas the probe graph problem on chordal graphs has polynomial-time complexity.[5] The set of all perfect elimination orderings of a chordal graph can be modeled as the basic words of an antimatroid; Chandran et al. (2003) use this connection to antimatroids as part of an algorithm for efficiently listing all perfect elimination orderings of a given chordal graph.

16.2 Maximal cliques and graph coloring

Another application of perfect elimination orderings is ﬁnding a maximum clique of a chordal graph in polynomial- time, while the same problem for general graphs is NP-complete. More generally, a chordal graph can have only linearly many maximal cliques, while non-chordal graphs may have exponentially many. To list all maximal cliques of a chordal graph, simply ﬁnd a perfect elimination ordering, form a clique for each vertex v together with the neighbors of v that are later than v in the perfect elimination ordering, and test whether each of the resulting cliques is maximal. The clique graphs of chordal graphs are the dually chordal graphs.[6] The largest maximal clique is a maximum clique, and, as chordal graphs are perfect, the size of this clique equals the chromatic number of the chordal graph. Chordal graphs are perfectly orderable: an optimal coloring may be obtained by applying a greedy coloring algorithm to the vertices in the reverse of a perfect elimination ordering.[7]

16.3 Minimal separators

In any graph, a vertex separator is a set of vertices the removal of which leaves the remaining graph disconnected; a separator is minimal if it has no proper subset that is also a separator. According to a theorem of Dirac (1961), chordal graphs are graphs in which each minimal separator is a clique; Dirac used this characterization to prove that chordal graphs are perfect. The family of chordal graphs may be deﬁned inductively as the graphs whose vertices can be divided into three nonempty subsets A, S, and B, such that A ∪ S and S ∪ B both form chordal induced subgraphs, S is a clique, and there are no edges from A to B. That is, they are the graphs that have a recursive decomposition by clique separators into smaller subgraphs. For this reason, chordal graphs have also sometimes been called decomposable graphs.[8]

16.4 Intersection graphs of subtrees

An alternative characterization of chordal graphs, due to Gavril (1974), involves trees and their subtrees. From a collection of subtrees of a tree, one can deﬁne a subtree graph, which is an intersection graph that has one vertex per subtree and an edge connecting any two subtrees that overlap in one or more nodes of the tree. Gavril showed that the subtree graphs are exactly the chordal graphs. 66 CHAPTER 16. CHORDAL GRAPH

A B F

C G

D E H

A B B F C G

C BB G E E C G E D E H

A chordal graph with eight vertices, represented as the intersection graph of eight subtrees of a six-node tree.

A representation of a chordal graph as an intersection of subtrees forms a tree decomposition of the graph, with treewidth equal to one less than the size of the largest clique in the graph; the tree decomposition of any graph G can be viewed in this way as a representation of G as a subgraph of a chordal graph. The tree decomposition of a graph is also the junction tree of the junction tree algorithm. 16.5. RELATION TO OTHER GRAPH CLASSES 67

16.5 Relation to other graph classes

16.5.1 Subclasses

Interval graphs are the intersection graphs of subtrees of path graphs, a special case of trees. Therefore, they are a subfamily of chordal graphs. Split graphs are graphs that are both chordal and the complements of chordal graphs. Bender, Richmond & Wormald (1985) showed that, in the limit as n goes to inﬁnity, the fraction of n-vertex chordal graphs that are split approaches one. Ptolemaic graphs are graphs that are both chordal and distance hereditary. Quasi-threshold graphs are a subclass of Ptolemaic graphs that are both chordal and cographs. Block graphs are another subclass of Ptolemaic graphs in which every two maximal cliques have at most one vertex in common. A special type is windmill graphs, where the common vertex is the same for every pair of cliques. Strongly chordal graphs are graphs that are chordal and contain no n-sun (for n ≥ 3) as an induced subgraph. Here an n-sun is an n-vertex chordal graph G together with a collection of n degree-two vertices, adjacent to the edges of a Hamiltonian cycle in G. K-trees are chordal graphs in which all maximal cliques and all maximal clique separators have the same size.[9] Apollonian networks are chordal maximal planar graphs, or equivalently planar 3-trees.[9] Maximal outerplanar graphs are a subclass of 2-trees, and therefore are also chordal.

16.5.2 Superclasses

Chordal graphs are a subclass of the well known perfect graphs. Other superclasses of chordal graphs include weakly chordal graphs, odd-hole-free graphs, and even-hole-free graphs. In fact, chordal graphs are precisely the graphs that are both odd-hole-free and even-hole-free (see holes in graph theory). Every chordal graph is a strangulated graph, a graph in which every peripheral cycle is a triangle, because peripheral cycles are a special case of induced cycles. Strangulated graphs are graphs that can be formed by clique-sums of chordal graphs and maximal planar graphs. Therefore strangulated graphs include maximal planar graphs.[10]

16.6 Chordal completions and treewidth

Main article: Chordal completion

If G is an arbitrary graph, a chordal completion of G (or minimum fill-in) is a chordal graph that contains G as a subgraph. The parameterized version of minimum fill-in is fixed parameter tractable, and moreover, is solvable in parameterized subexponential time.[11][12] The treewidth of G is one less than the number of vertices in a maximum clique of a chordal completion chosen to minimize this clique size. The k-trees are the graphs to which no additional edges can be added without increasing their treewidth to a number larger than k. Therefore, the k-trees are their own chordal completions, and form a subclass of the chordal graphs. Chordal completions can also be used to characterize several other related classes of graphs.[13]

16.7 Notes

[1] Dirac (1961).

[2] Weisstein, Eric W., “Triangulated Graph”, MathWorld.. Note however that “triangulated graphs” also sometimes refers to maximal planar graphs.

[3] Fulkerson & Gross (1965).

[4] Bodlaender, Fellows & Warnow (1992).

[5] Berry, Golumbic & Lipshteyn (2007). 68 CHAPTER 16. CHORDAL GRAPH

[6] Szwarcﬁter & Bornstein (1994).

[7] Maﬀray (2003).

[8] Peter Bartlett. “Undirected Graphical Models: Chordal Graphs, Decomposable Graphs, Junction Trees, and Factoriza- tions:" (PDF).

[9] Patil (1986).

[10] Seymour & Weaver (1984).

[11] Kaplan, Shamir & Tarjan (1999).

[12] Fomin & Villanger (2013).

[13] Parra & Scheﬄer (1997).

16.8 References

• Bender, E. A.; Richmond, L. B.; Wormald, N. C. (1985), “Almost all chordal graphs split”, J. Austral. Math. Soc.,A 38 (2): 214–221, doi:10.1017/S1446788700023077, MR 0770128.

• Berry, Anne; Golumbic, Martin Charles; Lipshteyn, Marina (2007), “Recognizing chordal probe graphs and cycle-bicolorable graphs”, SIAM Journal on Discrete Mathematics 21 (3): 573–591, doi:10.1137/050637091.

• Bodlaender, H. L.; Fellows, M. R.; Warnow, T. J. (1992), “Two strikes against perfect phylogeny”, Proc. of 19th International Colloquium on Automata Languages and Programming, Lecture Notes in Computer Science 623, pp. 273–283, doi:10.1007/3-540-55719-9_80.

• Chandran, L. S.; Ibarra, L.; Ruskey, F.; Sawada, J. (2003), “Enumerating and characterizing the perfect elimination orderings of a chordal graph” (PDF), Theoretical Computer Science 307 (2): 303–317, doi:10.1016/S0304- 3975(03)00221-4.

• Dirac, G. A. (1961), “On rigid circuit graphs”, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 25: 71–76, doi:10.1007/BF02992776, MR 0130190.

• Fomin, Fedor V.; Villanger, Yngve (2013), “Subexponential Parameterized Algorithm for Minimum Fill-In”, SIAM J. Comput. 6: 2197–2216, doi:10.1137/11085390X.

• Fulkerson, D. R.; Gross, O. A. (1965), “Incidence matrices and interval graphs”, Paciﬁc J. Math 15: 835–855, doi:10.2140/pjm.1965.15.835.

• Gavril, Fănică (1974), “The intersection graphs of subtrees in trees are exactly the chordal graphs”, Journal of Combinatorial Theory, Series B 16: 47–56, doi:10.1016/0095-8956(74)90094-X.

• Golumbic, Martin Charles (1980), Algorithmic Graph Theory and Perfect Graphs, Academic Press.

• Habib, Michel; McConnell, Ross; Paul, Christophe; Viennot, Laurent (2000), “Lex-BFS and partition re- ﬁnement, with applications to transitive orientation, interval graph recognition, and consecutive ones testing”, Theoretical Computer Science 234: 59–84, doi:10.1016/S0304-3975(97)00241-7.

• Kaplan, Haim; Shamir, Ron; Tarjan, Robert (1999), “Tractability of Parameterized Completion Problems on Chordal, Strongly Chordal, and Proper Interval Graphs”, SIAM J. Comput. 5: 1906–1922, doi:10.1137/S0097539796303044.

• Parra, Andreas; Scheﬄer, Petra (1997), “Characterizations and algorithmic applications of chordal graph embeddings”, Discrete Applied Mathematics 79 (1-3): 171–188, doi:10.1016/S0166-218X(97)00041-3, MR 1478250.

• Patil, H. P. (1986), “On the structure of k-trees”, Journal of Combinatorics, Information and System Sciences 11 (2–4): 57–64, MR 966069. 16.9. EXTERNAL LINKS 69

• Rose, D.; Lueker, George; Tarjan, Robert E. (1976), “Algorithmic aspects of vertex elimination on graphs”, SIAM Journal on Computing 5 (2): 266–283, doi:10.1137/0205021. • Seymour, P. D.; Weaver, R. W. (1984), “A generalization of chordal graphs”, Journal of Graph Theory 8 (2): 241–251, doi:10.1002/jgt.3190080206, MR 742878. • Swarcﬁter, J.L.; Bornstein, C.F. (1994), “Clique graphs of chordal and path graphs”, SIAM Journal on Discrete Mathematics 7: 331–336.

16.9 External links

• Information System on Graph Class Inclusions: chordal graph • Weisstein, Eric W., “Chordal Graph”, MathWorld. Chapter 17

Circulant graph

For the square matrices, see Circulant matrix. In graph theory, a circulant graph is an undirected graph that has a cyclic group of symmetries that includes a

12 0 1 11 2 10 3

9 4 8 5 7 6

The Paley graph of order 13, an example of a circulant graph. symmetry taking any vertex to any other vertex.

70 17.1. EQUIVALENT DEFINITIONS 71

17.1 Equivalent deﬁnitions

Circulant graphs can be described in several equivalent ways:[1]

• The automorphism group of the graph includes a cyclic subgroup that acts transitively on the graph’s vertices.

• The graph has an adjacency matrix that is a circulant matrix.

• The n vertices of the graph can be numbered from 0 to n − 1 in such a way that, if some two vertices numbered x and (x +d) mod n are adjacent, then every two vertices numbered z and (z +d) mod n are adjacent.

• The graph can be drawn (possibly with crossings) so that its vertices lie on the corners of a regular polygon, and every rotational symmetry of the polygon is also a symmetry of the drawing.

• The graph is a Cayley graph of a cyclic group.[2]

17.2 Examples

Every cycle graph is a circulant graph, as is every crown graph with 2 modulo 4 vertices. The Paley graphs of order n (where n is a prime number congruent to 1 modulo 4) is a graph in which the vertices are the numbers from 0 to n − 1 and two vertices are adjacent if their diﬀerence is a quadratic residue modulo n. Since the presence or absence of an edge depends only on the diﬀerence modulo n of two vertex numbers, any Paley graph is a circulant graph. Every Möbius ladder is a circulant graph, as is every complete graph.A complete bipartite graph is a circulant graph if it has the same number of vertices on both sides of its bipartition. If two numbers m and n are relatively prime, then the m × n rook’s graph (a graph that has a vertex for each square of an m × n chessboard and an edge for each two squares that a chess rook can move between in a single move) is a circulant graph. This is because its symmetries include as a subgroup the cyclic group {{{1}}}. More generally, in this case, the tensor product of graphs between any m- and n-vertex circulants is itself a circulant.[1] Many of the known lower bounds on Ramsey numbers come from examples of circulant graphs that have small maximum cliques and small maximum independent sets.[3]

17.3 A speciﬁc example

s1,...,sk − The circulant graph Cn with jumps s1, . . . , sk is deﬁned as the graph with n nodes labeled 0, 1, . . . , n 1 where each node i is adjacent to 2k nodes i s1, . . . , i sk mod n .

• s1,...,sk The graph Cn is connected if and only if gcd(n, s1, . . . , sk) = 1 .

• ≤ ··· s1,...,sk 2 If 1 s1 < < sk are ﬁxed integers then the number of spanning trees t(Cn ) = nan where an − satisﬁes a recurrence relation of order 2sk 1 .

• 1,2 2 In particular, t(Cn ) = nFn where Fn is the n-th Fibonacci number.

17.4 Self-complementary circulants

A self-complementary graph is a graph in which replacing every edge by a non-edge and vice versa produces an isomorphic graph. For instance, a ﬁve-vertex cycle graph is self-complementary, and is also a circulant graph. More generally every Paley graph is a self-complementary circulant graph.[4] Horst Sachs showed that, if a number n has the property that every prime factor of n is congruent to 1 modulo 4, then there exists a self-complementary circulant with n vertices. He conjectured that this condition is also necessary: that no other values of n allow a self-complementary circulant to exist.[1][4] The conjecture was proven some 40 years later, by Vilfred.[1] 72 CHAPTER 17. CIRCULANT GRAPH

17.5 Ádám’s conjecture

Deﬁne a circulant numbering of a circulant graph to be a labeling of the vertices of the graph by the numbers from 0 to n − 1 in such a way that, if some two vertices numbered x and y are adjacent, then every two vertices numbered z and (z − x + y) mod n are adjacent. Equivalently, a circulant numbering is a numbering of the vertices for which the adjacency matrix of the graph is a circulant matrix. Let a be an integer that is relatively prime to n, and let b be any integer. Then the linear function that takes a number x to ax + b transforms a circulant numbering to another circulant numbering. András Ádám conjectured that these linear maps are the only ways of renumbering a circulant graph while preserving the circulant property: that is, if G and H are isomorphic circulant graphs, with diﬀerent numberings, then there is a linear map that transforms the numbering for G into the numbering for H. However, Ádám’s conjecture is now known to be false. A counterexample is given by graphs G and H with 16 vertices each; a vertex x in G is connected to the six neighbors x ± 1, x ± 2, and x ± 7 modulo 16, while in H the six neighbors are x ± 2, x ± 3, and x ± 5 modulo 16. These two graphs are isomorphic, but their isomorphism cannot be realized by a linear map.[1]

17.6 Algorithmic questions

There is a polynomial-time recognition algorithm for circulant graphs, and the isomorphism problem for circulant graphs can be solved in polynomial time.[5]

17.7 References

[1] Vilfred, V. (2004), “On circulant graphs”, in Balakrishnan, R.; Sethuraman, G.; Wilson, Robin J., Graph Theory and its Applications (Anna University, Chennai, March 14–16, 2001), Alpha Science, pp. 34–36.

[2] Alspach, Brian (1997), “Isomorphism and Cayley graphs on abelian groups”, Graph symmetry (Montreal, PQ, 1996), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 497, Dordrecht: Kluwer Acad. Publ., pp. 1–22, MR 1468786.

[3] Small Ramsey Numbers, Stanisław P. Radziszowski, Electronic J. Combinatorics, dynamic survey, updated 2014.

[4] Sachs, Horst (1962). "Über selbstkomplementäre Graphen”. Publicationes Mathematicae Debrecen 9: 270–288. MR 0151953..

[5] Evdokimov, Sergei; Ponomarenko, Ilia (2004). “Recognition and veriﬁcation of an isomorphism of circulant graphs in polynomial time”. St. Petersburg Math. J. 15: 813–835. MR 2044629..

17.8 External links

• Weisstein, Eric W., “Circulant Graph”, MathWorld. Chapter 18

Claw-free graph

A claw

In graph theory, an area of mathematics, a claw-free graph is a graph that does not have a claw as an induced subgraph. A claw is another name for the complete bipartite graph K₁,₃ (that is, a star graph with three edges, three leaves, and one central vertex). A claw-free graph is a graph in which no induced subgraph is a claw; i.e., any subset of four vertices has other than only three edges connecting them in this pattern. Equivalently, a claw-free graph is a graph in which the neighborhood of any vertex is the complement of a triangle-free graph. Claw-free graphs were initially studied as a generalization of line graphs, and gained additional motivation through three key discoveries about them: the fact that all claw-free connected graphs of even order have perfect matchings, the discovery of polynomial time algorithms for ﬁnding maximum independent sets in claw-free graphs, and the characterization of claw-free perfect graphs.[1] They are the subject of hundreds of mathematical research papers and several surveys.[1]

73 74 CHAPTER 18. CLAW-FREE GRAPH

18.1 Examples

The regular icosahedron, a polyhedron whose vertices and edges form a claw-free graph.

• The line graph L(G) of any graph G is claw-free; L(G) has a vertex for every edge of G, and vertices are adjacent in L(G) whenever the corresponding edges share an endpoint in G. A line graph L(G) cannot contain a claw, because if three edges e1, e2, and e3 in G all share endpoints with another edge e4 then by the pigeonhole principle at least two of e1, e2, and e3 must share one of those endpoints with each other. Line graphs may be characterized in terms of nine forbidden subgraphs;[2] the claw is the simplest of these nine graphs. This characterization provided the initial motivation for studying claw-free graphs.[1] • The de Bruijn graphs (graphs whose vertices represent n-bit binary strings for some n, and whose edges represent (n − 1)-bit overlaps between two strings) are claw-free. One way to show this is via the construction of the de Bruijn graph for n-bit strings as the line graph of the de Bruijn graph for (n − 1)-bit strings. • The complement of any triangle-free graph is claw-free.[3] These graphs include as a special case any complete graph. • Proper interval graphs, the interval graphs formed as intersection graphs of families of intervals in which no interval contains another interval, are claw-free, because four properly intersecting intervals cannot intersect in the pattern of a claw.[3] • The Moser spindle, a seven-vertex graph used to provide a lower bound for the chromatic number of the plane, is claw-free. 18.2. RECOGNITION 75

• The graphs of several polyhedra and polytopes are claw-free, including the graph of the tetrahedron and more generally of any simplex (a complete graph), the graph of the octahedron and more generally of any cross polytope (isomorphic to the cocktail party graph formed by removing a perfect matching from a complete graph), the graph of the regular icosahedron,[4] and the graph of the 16-cell.

• The Schläﬂi graph, a strongly regular graph with 27 vertices, is claw-free.[4]

18.2 Recognition

It is straightforward to verify that a given graph with n vertices and m edges is claw-free in time O(n4), by testing each 4-tuple of vertices to determine whether they induce a claw.[5] Somewhat more eﬃciently, but more complicatedly, one can test whether a graph is claw-free by checking, for each vertex of the graph, that the complement graph of its neighbors does not contain a triangle. A graph contains a triangle if and only if the cube of its adjacency matrix contains a nonzero diagonal element, so ﬁnding a triangle may be performed in the same asymptotic time bound as n × n matrix multiplication.[6] Therefore, using the Coppersmith–Winograd algorithm, the total time for this claw-free recognition algorithm would be O(n3.376). Kloks, Kratsch & Müller (2000) observe that in any claw-free graph, each vertex has at most 2√m neighbors; for otherwise by Turán’s theorem the neighbors of the vertex would not have enough remaining edges to form the complement of a triangle-free graph. This observation allows the check of each neighborhood in the fast matrix multiplication based algorithm outlined above to be performed in the same asymptotic time bound as 2√m × 2√m matrix multiplication, or faster for vertices with even lower degrees. The worst case for this algorithm occurs when Ω(√m) vertices have Ω(√m) neighbors each, and the remaining vertices have few neighbors, so its total time is O(m3.376/2) = O(m1.688).

18.3 Enumeration

Because claw-free graphs include complements of triangle-free graphs, the number of claw-free graphs on n vertices grows at least as quickly as the number of triangle-free graphs, exponentially in the square of n. The numbers of connected claw-free graphs on n nodes, for n = 1, 2, ... are

1, 1, 2, 5, 14, 50, 191, 881, 4494, 26389, 184749, ... (sequence A022562 in OEIS).

If the graphs are allowed to be disconnected, the numbers of graphs are even larger: they are

1, 2, 4, 10, 26, 85, 302, 1285, 6170, ... (sequence A086991 in OEIS).

A technique of Palmer, Read & Robinson (2002) allows the number of claw-free cubic graphs to be counted very eﬃciently, unusually for graph enumeration problems.

18.4 Matchings

Sumner (1974) and, independently, Las Vergnas (1975) proved that every claw-free connected graph with an even number of vertices has a perfect matching.[7] That is, there exists a set of edges in the graph such that each vertex is an endpoint of exactly one of the matched edges. The special case of this result for line graphs implies that, in any graph with an even number of edges, one can partition the edges into paths of length two. Perfect matchings may be used to provide another characterization of the claw-free graphs: they are exactly the graphs in which every connected induced subgraph of even order has a perfect matching.[7] Sumner’s proof shows, more strongly, that in any connected claw-free graph one can ﬁnd a pair of adjacent vertices the removal of which leaves the remaining graph connected. To show this, Sumner ﬁnds a pair of vertices u and v that are as far apart as possible in the graph, and chooses w to be a neighbor of v that is as far from u as possible; as he shows, neither v nor w can lie on any shortest path from any other node to u, so the removal of v and w leaves the remaining graph connected. Repeatedly removing matched pairs of vertices in this way forms a perfect matching in the given claw-free graph. 76 CHAPTER 18. CLAW-FREE GRAPH

v u w

Sumner’s proof that claw-free connected graphs of even order have perfect matchings: removing the two adjacent vertices v and w that are farthest from u leaves a connected subgraph, within which the same removal process may be repeated.

The same proof idea holds more generally if u is any vertex, v is any vertex that is maximally far from u, and w is any neighbor of v that is maximally far from u. Further, the removal of v and w from the graph does not change any of the other distances from u. Therefore, the process of forming a matching by finding and removing pairs vw that are maximally far from u may be performed by a single postorder traversal of a breadth first search tree of the graph, rooted at u, in linear time. Chrobak, Naor & Novick (1989) provide an alternative linear-time algorithm based on depth-first search, as well as efficient parallel algorithms for the same problem. Faudree, Flandrin & Ryjáček (1997) list several related results, including the following: (r − 1)-connected K₁,r-free graphs of even order have perfect matchings for any r ≥ 2; claw-free graphs of odd order with at most one degree-one vertex may be partitioned into an odd cycle and a matching; for any k that is at most half the minimum degree of a claw-free graph in which either k or the number of vertices is even, the graph has a k-factor; and, if a claw-free graph is (2k + 1)-connected, then any k-edge matching can be extended to a perfect matching.

18.5 Independent sets

An independent set in a line graph corresponds to a matching in its underlying graph, a set of edges no two of which share an endpoint. The blossom algorithm of Edmonds (1965) finds a maximum matching in any graph in polynomial time, which is equivalent to computing a maximum independent set in line graphs. This has been independently extended to an algorithm for all claw-free graphs by Sbihi (1980) and Minty (1980).[8] Both approaches use the observation that in claw-free graphs, no vertex can have more than two neighbors in an independent set, and so the symmetric difference of two independent sets must induce a subgraph of degree at most two; that is, it is a union of paths and cycles. In particular, if I is a non-maximum independent set, it differs from any maximum independent set by even cycles and so called augmenting paths: induced paths which alternate between vertices not in I and vertices in I, and for which both endpoints have only one neighbor in I. As the symmetric difference of I with any augmenting path gives a larger independent set, the task thus reduces to searching for augmenting paths until no more can be found, analogously as in algorithms for finding maximum matchings. Sbihi’s algorithm recreates the blossom contraction step of Edmonds’ algorithm and adds a similar, but more complicated, clique contraction step. Minty’s approach is to transform the problem instance into an auxiliary line graph and use Edmonds’ algorithm directly to find the augmenting paths. After a correction by Nakamura & Tamura 2001, 18.6. COLORING, CLIQUES, AND DOMINATION 77

A non-maximum independent set (the two violet nodes) and an augmenting path

Minty’s result may also be used to solve in polynomial time the more general problem of ﬁnding in claw-free graphs an independent set of maximum weight. Generalizations of these results to wider classes of graphs are also known.[8] By showing a novel structure theorem, Faenza, Oriolo & Stauﬀer (2011) gave a cubic time algorithm, which also works in the weighted setting.

18.6 Coloring, cliques, and domination

A perfect graph is a graph in which the chromatic number and the size of the maximum clique are equal, and in which this equality persists in every induced subgraph. It is now known (the strong perfect graph theorem) that perfect graphs may be characterized as the graphs that do not have as induced subgraphs either an odd cycle or the complement of an odd cycle (a so-called odd hole).[9] However, for many years this remained an unsolved conjecture, only proven for special subclasses of graphs. One of these subclasses was the family of claw-free graphs: it was discovered by several authors that claw-free graphs without odd cycles and odd holes are perfect. Perfect claw-free graphs may be recognized in polynomial time. In a perfect claw-free graph, the neighborhood of any vertex forms the complement of a bipartite graph. It is possible to color perfect claw-free graphs, or to find maximum cliques in them, in polynomial time.[10] In general, it is NP-hard to find the largest clique in a claw-free graph.[5] It is also NP-hard to find an optimal coloring of the graph, because (via line graphs) this problem generalizes the NP-hard problem of computing the chromatic index of a graph.[5] For the same reason, it is NP-hard to find a coloring that achieves an approximation ratio better than 4/3. However, an approximation ratio of two can be achieved by a greedy coloring algorithm, because the chromatic number of a claw-free graph is greater than half its maximum degree. Although not every claw-free graph is perfect, claw-free graphs satisfy another property, related to perfection. A graph is called domination perfect if it has a minimum dominating set that is independent, and if the same property holds in all of its induced subgraphs. Claw-free graphs have this property. To see this, let D be a dominating set in a claw-free graph, and suppose that v and w are two adjacent vertices in D; then the set of vertices dominated by v but not by w must be a clique (else v would be the center of a claw). If every vertex in this clique is already 78 CHAPTER 18. CLAW-FREE GRAPH

dominated by at least one other member of D, then v can be removed producing a smaller independent dominating set, and otherwise v can be replaced by one of the undominated vertices in its clique producing a dominating set with fewer adjacencies. By repeating this replacement process one eventually reaches a dominating set no larger than D, so in particular when the starting set D is a minimum dominating set this process forms an equally small independent dominating set.[11] Despite this domination perfectness property, it is NP-hard to determine the size of the minimum dominating set in a claw-free graph.[5] However, in contrast to the situation for more general classes of graphs, ﬁnding the minimum dominating set or the minimum connected dominating set in a claw-free graph is ﬁxed-parameter tractable: it can be solved in time bounded by a polynomial in the size of the graph multiplied by an exponential function of the dominating set size.[12]

18.7 Structure

Chudnovsky & Seymour (2005) overview a series of papers in which they prove a structure theory for claw-free graphs, analogous to the graph structure theorem for minor-closed graph families proven by Robertson and Seymour, and to the structure theory for perfect graphs that Chudnovsky, Seymour and their co-authors used to prove the strong perfect graph theorem.[9] The theory is too complex to describe in detail here, but to give a ﬂavor of it, it suﬃces to outline two of their results. First, for a special subclass of claw-free graphs which they call quasi-line graphs (equivalently, locally co-bipartite graphs), they state that every such graph has one of two forms:

1. A fuzzy circular interval graph, a class of graphs represented geometrically by points and arcs on a circle, generalizing proper circular arc graphs.

2. A graph constructed from a multigraph by replacing each edge by a fuzzy linear interval graph. This generalizes the construction of a line graph, in which every edge of the multigraph is replaced by a vertex. Fuzzy linear interval graphs are constructed in the same way as fuzzy circular interval graphs, but on a line rather than on a circle.

Chudnovsky and Seymour classify arbitrary connected claw-free graphs into one of the following:

1. Six speciﬁc subclasses of claw-free graphs. Three of these are line graphs, proper circular arc graphs, and the induced subgraphs of an icosahedron; the other three involve additional deﬁnitions.

2. Graphs formed in four simple ways from smaller claw-free graphs.

3. Antiprismatic graphs, a class of dense graphs deﬁned as the claw-free graphs in which every four vertices induce a subgraph with at least two edges.

Much of the work in their structure theory involves a further analysis of antiprismatic graphs. The Schläﬂi graph, a claw-free strongly regular graph with parameters srg(27,16,10,8), plays an important role in this part of the analysis. This structure theory has led to new advances in polyhedral combinatorics and new bounds on the chromatic number of claw-free graphs,[4][13] as well as to new ﬁxed-parameter-tractable algorithms for dominating sets in claw-free graphs.[14]

18.8 Notes

[1] Faudree, Flandrin & Ryjáček (1997), p. 88.

[2] Beineke (1968).

[3] Faudree, Flandrin & Ryjáček (1997), p. 89.

[4] Chudnovsky & Seymour (2005).

[5] Faudree, Flandrin & Ryjáček (1997), p. 132.

[6] Itai & Rodeh (1978). 18.9. REFERENCES 79

[7] Faudree, Flandrin & Ryjáček (1997), pp. 120–124.

[8] Faudree, Flandrin & Ryjáček (1997), pp. 133–134.

[9] Chudnovsky et al. (2006).

[10] Faudree, Flandrin & Ryjáček (1997), pp. 135–136.

[11] Faudree, Flandrin & Ryjáček (1997), pp. 124–125.

[12] Cygan et al. (2011); Hermelin et al. (2010).

[13] King & Reed (2015).

[14] Hermelin et al. (2010).

18.9 References

• Beineke, L. W. (1968), “Derived graphs of digraphs”, in Sachs, H.; Voss, H.-J.; Walter, H.-J., Beiträge zur Graphentheorie, Leipzig: Teubner, pp. 17–33. • Chrobak, Marek; Naor, Joseph; Novick, Mark B. (1989), “Using bounded degree spanning trees in the design of efficient algorithms on claw-free graphs”, in Dehne, F.; Sack, J.-R.; Santoro, N., Algorithms and Data Structures: Workshop WADS '89, Ottawa, Canada, August 1989, Proceedings, Lecture Notes in Comput. Sci. 382, Berlin: Springer, pp. 147–162, doi:10.1007/3-540-51542-9_13. • Chudnovsky, Maria; Robertson, Neil; Seymour, Paul; Thomas, Robin (2006), “The strong perfect graph theorem” (PDF), Annals of Mathematics 164 (1): 51–229, doi:10.4007/annals.2006.164.51. • Chudnovsky, Maria; Seymour, Paul (2005), “The structure of claw-free graphs” (PDF), Surveys in combinatorics 2005, London Math. Soc. Lecture Note Ser. 327, Cambridge: Cambridge Univ. Press, pp. 153–171, MR 2187738. • Cygan, Marek; Philip, Geevarghese; Pilipczuk, Marcin; Pilipczuk, Michał; Wojtaszczyk, Jakub Onufry (2011), “Dominating set is fixed parameter tractable in claw-free graphs”, Theoretical Computer Science 412 (50): 6982–7000, arXiv:1011.6239, doi:10.1016/j.tcs.2011.09.010, MR 2894366. • Edmonds, Jack (1965), “Paths, Trees and Flowers”, Canadian J. Math 17 (0): 449–467, doi:10.4153/CJM- 1965-045-4, MR 0177907. • Faenza, Yuri; Oriolo, Gianpaolo; Stauffer, Gautier (2011), “An Algorithmic Decomposition of Claw-free Graphs Leading to an O(n3)-algorithm for the Weighted Stable Set Problem”, Proceedings of the Twenty- second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '11 (3–4), San Francisco, California: SIAM, pp. 630–646, doi:10.1137/1.9781611973082.49. • Faudree, Ralph; Flandrin, Evelyne; Ryjáček, Zdeněk (1997), “Claw-free graphs — A survey”, Discrete Math- ematics 164 (1–3): 87–147, doi:10.1016/S0012-365X(96)00045-3, MR 1432221. • Goldberg, Andrew V.; Karzanov, Alexander V. (1996), “Path problems in skew-symmetric graphs”, Combi- natorica 16 (3): 353–382, doi:10.1007/BF01261321. • Hermelin, Danny; Mnich, Matthias; van Leeuwen, Erik Jan; Woeginger, Gerhard (2010), Domination when the stars are out, arXiv:1012.0012. • Itai, Alon; Rodeh, Michael (1978), “Finding a minimum circuit in a graph”, SIAM Journal on Computing 7 (4): 413–423, doi:10.1137/0207033, MR 0508603. • King, Andrew D.; Reed, Bruce A. (2015), “Claw-free graphs, skeletal graphs, and a stronger conjecture on ω, Δ, and χ", Journal of Graph Theory 78 (3): 157–194, arXiv:1212.3036, doi:10.1002/jgt.21797. • Kloks, Ton; Kratsch, Dieter; Müller, Haiko (2000), “Finding and counting small induced subgraphs efficiently”, Information Processing Letters 74 (3–4): 115–121, doi:10.1016/S0020-0190(00)00047-8, MR 1761552. • Las Vergnas, M. (1975), “A note on matchings in graphs”, Cahiers du Centre d'Études de Recherche Opéra- tionnelle 17 (2-3-4): 257–260, MR 0412042. 80 CHAPTER 18. CLAW-FREE GRAPH

• Minty, George J. (1980), “On maximal independent sets of vertices in claw-free graphs”, Journal of Combi- natorial Theory. Series B 28 (3): 284–304, doi:10.1016/0095-8956(80)90074-X, MR 579076. • Nakamura, Daishin; Tamura, Akihisa (2001), “A revision of Minty’s algorithm for ﬁnding a maximum weighted stable set of a claw-free graph”, Journal of the Operations Research Society of Japan 44 (2): 194–204. • Palmer, Edgar M.; Read, Ronald C.; Robinson, Robert W. (2002), “Counting claw-free cubic graphs” (PDF), SIAM Journal on Discrete Mathematics 16 (1): 65–73, doi:10.1137/S0895480194274777, MR 1972075. • Sbihi, Najiba (1980), “Algorithme de recherche d'un stable de cardinalité maximum dans un graphe sans étoile”, Discrete Mathematics 29 (1): 53–76, doi:10.1016/0012-365X(90)90287-R, MR 553650. • Sumner, David P. (1974), “Graphs with 1-factors”, Proceedings of the American Mathematical Society (Amer- ican Mathematical Society) 42 (1): 8–12, doi:10.2307/2039666, JSTOR 2039666, MR 0323648.

18.10 External links

• Claw-free graphs, Information System on Graph Class Inclusions • Mugan, Jonathan William; Weisstein, Eric W., “Claw-Free Graph”, MathWorld. Chapter 19

Cograph

The Turán graph T(13,4), an example of a cograph

In graph theory, a cograph, or complement-reducible graph, or P4-free graph, is a graph that can be generated from the single-vertex graph K1 by complementation and disjoint union. That is, the family of cographs is the smallest class of graphs that includes K1 and is closed under complementation and disjoint union.

81 82 CHAPTER 19. COGRAPH

Cographs have been discovered independently by several authors since the 1970s; early references include Jung (1978), Lerchs (1971), Seinsche (1974), and Sumner (1974). They have also been called D*-graphs,[1] hereditary Dacey graphs (after the related work of James C. Dacey, Jr. on orthomodular lattices),[2] and 2-parity graphs.[3] They have a simple structural decomposition involving disjoint union and complement graph operations that can be represented concisely by a labeled tree, and used algorithmically to eﬃciently solve many problems such as ﬁnding the maximum clique that are hard on more general graph classes. Special cases of the cographs include the complete graphs, complete bipartite graphs, and threshold graphs. The cographs are, in turn, special cases of the distance-hereditary graphs, comparability graphs, and perfect graphs.

19.1 Deﬁnition

19.1.1 Recursive construction

Any cograph may be constructed using the following rules:

1. any single vertex graph is a cograph;

2. if G is a cograph, so is its complement graph G ;

3. if G and H are cographs, so is their disjoint union G ∪ H .

The cographs may be deﬁned as the graphs that can be constructed using these operations, starting from the single- vertex graphs.[4] Alternatively, instead of using the complement operation, one can use the join operation, which consists of forming the disjoint union G ∪ H and then adding an edge between every pair of a vertex from G and a vertex from H .

19.1.2 Other characterizations

Several alternative characterizations of cographs can be given. Among them:

• A cograph is a graph which does not contain the path P4 on 4 vertices (and hence of length 3) as an induced subgraph. That is, a graph is a cograph if and only if for any four vertices v1, v2, v3, v4 , if {v1, v2}, {v2, v3} [4] and {v3, v4} are edges of the graph then at least one of {v1, v3}, {v1, v4} or {v2, v4} is also an edge.

• A cograph is a graph all of whose induced subgraphs have the property that any maximal clique intersects any maximal independent set in a single vertex.

• A cograph is a graph in which every nontrivial induced subgraph has at least two vertices with the same neigh- bourhoods.

• A cograph is a graph in which every connected induced subgraph has a disconnected complement.

• A cograph is a graph all of whose connected induced subgraphs have diameter at most 2.

• A cograph is a graph in which every connected component is a distance-hereditary graph with diameter at most 2.

• A cograph is a graph with clique-width at most 2.[5]

• A cograph is a comparability graph of a series-parallel partial order.[1]

• A cograph is a permutation graph of a separable permutation.[6]

• A cograph is a graph all of whose minimal chordal completions are trivially perfect graphs.[7] 19.2. COTREES 83

1 a

0 0 b g

a 1 1 f g c f

b c d e d e

A cotree and the corresponding cograph. Each edge (u,v) in the cograph has a matching color to the least common ancestor of u and v in the cotree.

19.2 Cotrees

A cotree is a tree in which the internal nodes are labeled with the numbers 0 and 1. Every cotree T deﬁnes a cograph G having the leaves of T as vertices, and in which the subtree rooted at each node of T corresponds to the induced subgraph in G deﬁned by the set of leaves descending from that node:

• A subtree consisting of a single leaf node corresponds to an induced subgraph with a single vertex.

• A subtree rooted at a node labeled 0 corresponds to the union of the subgraphs deﬁned by the children of that node.

• A subtree rooted at a node labeled 1 corresponds to the join of the subgraphs deﬁned by the children of that node; that is, we form the union and add an edge between every two vertices corresponding to leaves in diﬀerent subtrees. Alternatively, the join of a set of graphs can be viewed as formed by complementing each graph, forming the union of the complements, and then complementing the resulting union.

An equivalent way of describing the cograph formed from a cotree is that two vertices are connected by an edge if and only if the lowest common ancestor of the corresponding leaves is labeled by 1. Conversely, every cograph can be represented in this way by a cotree. If we require the labels on any root-leaf path of this tree to alternate between 0 and 1, this representation is unique.[4]

19.3 Computational properties

Cographs may be recognized in linear time, and a cotree representation constructed, using modular decomposition,[8] partition reﬁnement,[9] LexBFS ,[10] or split decomposition.[11] Once a cotree representation has been constructed, many familiar graph problems may be solved via simple bottom-up calculations on the cotrees. For instance, to ﬁnd the maximum clique in a cograph, compute in bottom-up order the maximum clique in each subgraph represented by a subtree of the cotree. For a node labeled 0, the maximum clique is the maximum among the cliques computed for that node’s children. For a node labeled 1, the maximum clique is the union of the cliques computed for that node’s children, and has size equal to the sum of the children’s clique sizes. Thus, by alternately maximizing and summing values stored at each node of the cotree, we may compute the maximum clique size, and by alternately maximizing and taking unions, we may construct the maximum clique itself. Similar bottom-up tree computations allow the maximum independent set, vertex coloring number, maximum clique cover, and Hamiltonicity (that is the existence of a Hamiltonian cycle) to be computed in linear time from a cotree representation of a cograph.[4] Two cographs are isomorphic if and only if their cotrees (in the canonical form with no two adjacent vertices with the same label) are isomorphic. Because of this equivalence, one can determine in linear time whether two cographs are isomorphic, by constructing their cotrees and applying a linear time isomorphism test for labeled trees. [4] 84 CHAPTER 19. COGRAPH

If H is an induced subgraph of a cograph G, then H is itself a cograph; the cotree for H may be formed by removing some of the leaves from the cotree for G and then suppressing nodes that have only one child. It follows from Kruskal’s tree theorem that the relation of being an induced subgraph is a well-quasi-ordering on the cographs.[12] Thus, if a subfamily of the cographs (such as the planar cographs) is closed under induced subgraph operations then it has a ﬁnite number of forbidden induced subgraphs. Computationally, this means that testing membership in such a subfamily may be performed in linear time, by using a bottom-up computation on the cotree of a given graph to test whether it contains any of these forbidden subgraphs. However, when the sizes of two cographs are both variable, testing whether one of them is an induced subgraph of the other is NP-complete.[13]

19.4 Related graph families

19.4.1 Subclasses

Every complete graph Kn is a cograph, with a cotree consisting of a single 1-node and n leaves. Similarly, every complete bipartite graph Ka,b is a cograph. Its cotree is rooted at a 1-node which has two 0-node children, one with a leaf children and one with b leaf children. A Turán graph may be formed by the join of a family of equal- sized independent sets; thus, it also is a cograph, with a cotree rooted at a 1-node that has a child 0-node for each independent set. Every threshold graph is also a cograph. A threshold graph may be formed by repeatedly adding one vertex, either connected to all previous vertices or to none of them; each such operation is one of the disjoint union or join operations by which a cotree may be formed. [14]

19.4.2 Superclasses

The characterization of cographs by the property that every clique and maximal independent set have a nonempty intersection is a stronger version of the deﬁning property of strongly perfect graphs, in which there every induced subgraph contains an independent set that intersects all maximal cliques. In a cograph, every maximal independent set intersects all maximal cliques. Thus, every cograph is strongly perfect.[15] Every cograph is a distance-hereditary graph, meaning that every induced path in a cograph is a shortest path. The cographs may be characterized among the distance-hereditary graphs as having diameter two in each connected component. Every cograph is also a comparability graph of a series-parallel partial order, obtained by replacing the disjoint union and join operations by which the cograph was constructed by disjoint union and ordinal sum operations on partial orders. Because strongly perfect graphs, distance-hereditary graphs, and comparability graphs are all perfect graphs, cographs are also perfect.[14]

19.5 Notes

[1] Jung (1978).

[2] Sumner (1974).

[3] Burlet & Uhry (1984).

[4] Corneil, Lerchs & Stewart Burlingham (1981).

[5] Courcelle & Olariu (2000).

[6] Bose, Buss & Lubiw (1998).

[7] Parra & Scheﬄer (1997).

[8] Corneil, Perl & Stewart (1985).

[9] Habib & Paul (2005).

[10] Bretscher et al. (2008).

[11] Gioan & Paul (2012). 19.6. REFERENCES 85

[12] Damaschke (1990).

[13] Damaschke (1991).

[14] Brandstädt, Le & Spinrad (1999).

[15] Berge & Duchet (1984).

19.6 References

• Berge, C.; Duchet, P. (1984), “Strongly perfect graphs”, Topics on Perfect Graphs, North-Holland Mathematics Studies 88, Amsterdam: North-Holland, pp. 57–61, doi:10.1016/S0304-0208(08)72922-0, MR 778749.

• Bose, Prosenjit; Buss, Jonathan; Lubiw, Anna (1998), “Pattern matching for permutations”, Information Pro- cessing Letters 65: 277–283, doi:10.1016/S0020-0190(97)00209-3, MR 1620935.

• Brandstädt, Andreas; Le, Van Bang; Spinrad, Jeremy P. (1999), Graph Classes: A Survey, SIAM Monographs on Discrete Mathematics and Applications, ISBN 978-0-89871-432-6.

• Burlet, M.; Uhry, J. P. (1984), “Parity Graphs”, Topics on Perfect Graphs, Annals of Discrete Mathematics 21, pp. 253–277.

• Bretscher, A.; Corneil, D. G.; Habib, M.; Paul, C. (2008), “A simple Linear Time LexBFS Cograph Recogni- tion Algorithm”, SIAM Journal on Discrete Mathematics 22 (4): 1277–1296.

• Corneil, D. G.; Lerchs, H.; Stewart Burlingham, L. (1981), “Complement reducible graphs”, Discrete Applied Mathematics 3 (3): 163–174, doi:10.1016/0166-218X(81)90013-5, MR 0619603.

• Corneil, D. G.; Perl, Y.; Stewart, L. K. (1985), “A linear recognition algorithm for cographs”, SIAM Journal on Computing 14 (4): 926–934, doi:10.1137/0214065, MR 0807891.

• Courcelle, B.; Olariu, S. (2000), “Upper bounds to the clique width of graphs”, Discrete Applied Mathematics 101 (1–3): 77–144, doi:10.1016/S0166-218X(99)00184-5, MR 1743732.

• Damaschke, Peter (1990), “Induced subgraphs and well-quasi-ordering”, Journal of Graph Theory 14 (4): 427–435, doi:10.1002/jgt.3190140406, MR 1067237.

• Damaschke, Peter (1991), “Induced subraph isomorphism for cographs is NP-complete”, in Möhring, Rolf H., Graph-Theoretic Concepts in Computer Science: 16th International Workshop WG '90 Berlin, Germany, June 20–22, 1990, Proceedings, Lecture Notes in Computer Science 484, Springer-Verlag, pp. 72–78, doi:10.1007/3- 540-53832-1_32.

• Gioan, Emeric; Paul, Christophe (2012), “Split decomposition and graph-labelled trees: characterizations and fully dynamic algorithms for totally decomposable graphs”, Discrete Applied Mathematics 160 (6): 708–733, arXiv:0810.1823, doi:10.1016/j.dam.2011.05.007, MR 2901084.

• Habib, Michel; Paul, Christophe (2005), “A simple linear time algorithm for cograph recognition” (PDF), Discrete Applied Mathematics 145 (2): 183–197, doi:10.1016/j.dam.2004.01.011, MR 2113140.

• Jung, H. A. (1978), “On a class of posets and the corresponding comparability graphs”, Journal of Combina- torial Theory, Series B 24 (2): 125–133, doi:10.1016/0095-8956(78)90013-8, MR 0491356.

• Lerchs, H. (1971), On cliques and kernels, Tech. Report, Dept. of Comp. Sci., Univ. of Toronto.

• Parra, Andreas; Scheﬄer, Petra (1997), “Characterizations and algorithmic applications of chordal graph embeddings”, 4th Twente Workshop on Graphs and Combinatorial Optimization (Enschede, 1995), Discrete Ap- plied Mathematics 79 (1-3): 171–188, doi:10.1016/S0166-218X(97)00041-3, MR 1478250.

• Seinsche, D. (1974), “On a property of the class of n-colorable graphs”, Journal of Combinatorial Theory, Series B 16 (2): 191–193, doi:10.1016/0095-8956(74)90063-X, MR 0337679.

• Sumner, D. P. (1974), “Dacey graphs”, Journal of the Australian Mathematical Society 18 (4): 492–502, doi:10.1017/S1446788700029232, MR 0382082. 86 CHAPTER 19. COGRAPH

19.7 External links

• “Cograph graphs”, Information System on Graph Class Inclusions

• Weisstein, Eric W., “Cograph”, MathWorld. Chapter 20

Comparability graph

In graph theory, a comparability graph is an undirected graph that connects pairs of elements that are comparable to each other in a partial order. Comparability graphs have also been called transitively orientable graphs, partially orderable graphs, and containment graphs.[1] An incomparability graph is an undirected graph that connects pairs of elements that are not comparable to each other in a partial order.

20.1 Deﬁnitions and characterization

A A D

C D C B

Hasse diagram of a poset (left) and its comparability graph (right)

For any strict partially ordered set (S,<), the comparability graph of (S, <) is the graph (S, ⊥) of which the vertices are the elements of S and the edges are those pairs {u, v} of elements such that u < v. That is, for a partially ordered set, take the directed acyclic graph, apply transitive closure, and remove orientation. Equivalently, a comparability graph is a graph that has a transitive orientation,[2] an assignment of directions to the edges of the graph (i.e. an orientation of the graph) such that the adjacency relation of the resulting directed graph is transitive: whenever there exist directed edges (x,y) and (y,z), there must exist an edge (x,z). One can represent any partial order as a family of sets, such that x < y in the partial order whenever the set corresponding to x is a subset of the set corresponding to y. In this way, comparability graphs can be shown to be equivalent to containment graphs of set families; that is, a graph with a vertex for each set in the family and an edge between two sets whenever one is a subset of the other.[3] Alternatively,[4] a comparability graph is a graph such that, for every generalized cycle of odd length, one can ﬁnd an edge (x,y) connecting two vertices that are at distance two in the cycle. Such an edge is called a triangular chord. In this context, a generalized cycle is deﬁned to be a closed walk that uses each edge of the graph at most once in each direction. Comparability graphs can also be characterized by a list of forbidden induced subgraphs.[5]

87 88 CHAPTER 20. COMPARABILITY GRAPH a

c d

e f

One of the forbidden induced subgraphs of a comparability graph. The generalized cycle a–b–d–f–d–c–e–c–b–a in this graph has odd length (nine) but has no triangular chords.

20.2 Relation to other graph families

Every complete graph is a comparability graph, the comparability graph of a total order. All acyclic orientations of a complete graph are transitive. Every bipartite graph is also a comparability graph. Orienting the edges of a bipartite graph from one side of the bipartition to the other results in a transitive orientation, corresponding to a partial order of height two. As Seymour (2006) observes, every comparability graph that is neither complete nor bipartite has a skew partition. The complement of any interval graph is a comparability graph. The comparability relation is called an interval order. Interval graphs are exactly the graphs that are chordal and that have comparability graph complements.[6] A permutation graph is a containment graph on a set of intervals.[7] Therefore, permutation graphs are another subclass of comparability graphs. The trivially perfect graphs are the comparability graphs of rooted trees.[8] Cographs can be characterized as the comparability graphs of series-parallel partial orders; thus, cographs are also comparability graphs.[9] Threshold graphs are another special kind of comparability graph. Every comparability graph is perfect. The perfection of comparability graphs is Mirsky’s theorem, and the perfection of their complements is Dilworth’s theorem; these facts, together with the complementation property of perfect graphs can be used to prove Dilworth’s theorem from Mirsky’s theorem or vice versa.[10] More speciﬁcally, comparability graphs are perfectly orderable graphs, a subclass of perfect graphs: a greedy coloring algorithm for a topological ordering of a transitive orientation of the graph will optimally color them.[11] 20.3. ALGORITHMS 89

The complement of every comparability graph is a string graph.[12]

20.3 Algorithms

A transitive orientation of a graph, if it exists, can be found in linear time.[13] However, the algorithm for doing so will assign orientations to the edges of any graph, so to complete the task of testing whether a graph is a comparability graph, one must test whether the resulting orientation is transitive, a problem provably equivalent in complexity to matrix multiplication. Because comparability graphs are perfect, many problems that are hard on more general classes of graphs, including graph coloring and the independent set problem, can be computed in polynomial time for comparability graphs.

20.4 Notes

[1] Golumbic (1980), p. 105; Brandstädt, Le & Spinrad (1999), p. 94.

[2] Ghouila-Houri (1962); see Brandstädt, Le & Spinrad (1999), theorem 1.4.1, p. 12. Although the orientations coming from partial orders are acyclic, it is not necessary to include acyclicity as a condition of this characterization.

[3] Urrutia (1989); Trotter (1992); Brandstädt, Le & Spinrad (1999), section 6.3, pp. 94–96.

[4] Ghouila-Houri (1962) and Gilmore & Hoﬀman (1964). See also Brandstädt, Le & Spinrad (1999), theorem 6.1.1, p. 91.

[5] Gallai (1967); Trotter (1992); Brandstädt, Le & Spinrad (1999), p. 91 and p. 112.

[6] Transitive orientability of interval graph complements was proven by Ghouila-Houri (1962); the characterization of interval graphs is due to Gilmore & Hoﬀman (1964). See also Golumbic (1980), prop. 1.3, pp. 15–16.

[7] Dushnik & Miller (1941). Brandstädt, Le & Spinrad (1999), theorem 6.3.1, p. 95.

[8] Brandstädt, Le & Spinrad (1999), theorem 6.6.1, p. 99.

[9] Brandstädt, Le & Spinrad (1999), corollary 6.4.1, p. 96; Jung (1978).

[10] Golumbic (1980), theorems 5.34 and 5.35, p. 133.

[11] Maﬀray (2003).

[12] Golumbic, Rotem & Urrutia (1983) and Lovász (1983). See also Fox & Pach (2012).

[13] McConnell & Spinrad (1997); see Brandstädt, Le & Spinrad (1999), p. 91.

20.5 References

• Brandstädt, Andreas; Le, Van Bang; Spinrad, Jeremy (1999), Graph Classes: A Survey, SIAM Monographs on Discrete Mathematics and Applications, ISBN 0-89871-432-X. • Dushnik, Ben; Miller, E. W. (1941), “Partially ordered sets”, American Journal of Mathematics (The Johns Hopkins University Press) 63 (3): 600–610, doi:10.2307/2371374, JSTOR 2371374, MR 0004862. • Fox, J.; Pach, J. (2012), “String graphs and incomparability graphs” (PDF), Advances in Mathematics 230 (3): 1381–1401, doi:10.1016/j.aim.2012.03.011. • Gallai, Tibor (1967), “Transitiv orientierbare Graphen”, Acta Math. Acad. Sci. Hung. 18: 25–66, doi:10.1007/BF02020961, MR 0221974. • Ghouila-Houri, Alain (1962), “Caractérisation des graphes non orientés dont on peut orienter les arrêtes de manière à obtenir le graphe d'une relation d'ordre”, Les Comptes rendus de l'Académie des sciences 254: 1370– 1371, MR 0172275. • Gilmore, P. C.; Hoﬀman, A. J. (1964), “A characterization of comparability graphs and of interval graphs”, Canadian Journal of Mathematics 16: 539–548, doi:10.4153/CJM-1964-055-5, MR 0175811. 90 CHAPTER 20. COMPARABILITY GRAPH

• Golumbic, Martin Charles (1980), Algorithmic Graph Theory and Perfect Graphs, Academic Press, ISBN 0- 12-289260-7. • Golumbic, M.; Rotem, D.; Urrutia, J. (1983), “Comparability graphs and intersection graphs”, Discrete Math- ematics 43 (1): 37–46, doi:10.1016/0012-365X(83)90019-5. • Jung, H. A. (1978), “On a class of posets and the corresponding comparability graphs”, Journal of Combina- torial Theory, Series B 24 (2): 125–133, doi:10.1016/0095-8956(78)90013-8, MR 0491356. • Lovász, L. (1983), “Perfect graphs”, Selected Topics in Graph Theory 2, London: Academic Press, pp. 55–87.

• Maﬀray, Frédéric (2003), “On the coloration of perfect graphs”, in Reed, Bruce A.; Sales, Cláudia L., Recent Advances in Algorithms and Combinatorics, CMS Books in Mathematics 11, Springer-Verlag, pp. 65–84, doi:10.1007/0-387-22444-0_3. • McConnell, R. M.; Spinrad, J. (1997), “Linear-time transitive orientation”, 8th ACM-SIAM Symposium on Discrete Algorithms, pp. 19–25.

• Seymour, Paul (2006), “How the proof of the strong perfect graph conjecture was found” (PDF), Gazette des Mathématiciens (109): 69–83, MR 2245898.

• Trotter, William T. (1992), Combinatorics and Partially Ordered Sets — Dimension Theory, Johns Hopkins University Press.

• Urrutia, Jorge (1989), Rival, I., ed., Partial orders and Euclidean geometry, Kluwer Academic Publishers, pp. 327–436. Chapter 21

Conference graph

In the mathematical area of graph theory, a conference graph is a strongly regular graph with parameters v, k = (v − 1)/2, λ = (v − 5)/4, and μ = (v − 1)/4. It is the graph associated with a symmetric conference matrix, and consequently its order v must be 1 (modulo 4) and a sum of two squares. Conference graphs are known to exist for all small values of v allowed by the restrictions, e.g., v = 5, 9, 13, 17, 25, 29, and (the Paley graphs) for all prime powers congruent to 1 (modulo 4). However, there are many values of v that are allowed, for which the existence of a conference graph is unknown. The eigenvalues of a conference graph need not be integers, unlike those of other strongly regular graphs. If the graph is connected, the eigenvalues are k with multiplicity 1, and two other eigenvalues,

√ −1 v , 2 each with multiplicity (v − 1)/2.

21.1 References

Brouwer, A.E., Cohen, A.M., and Neumaier, A. (1989), Distance Regular Graphs. Berlin, New York: Springer- Verlag. ISBN 3-540-50619-5, ISBN 0-387-50619-5

91 Chapter 22

Convex bipartite graph

In the mathematical field of graph theory, a convex bipartite graph is a bipartite graph with specific properties. A bipartite graph, (U ∪ V, E), is said to be convex over the vertex set U if U can be enumerated such that for all v ∈ V the vertices adjacent to v are consecutive. Convexity over V is defined analogously. A bipartite graph (U ∪ V, E) that is convex over both U and V is said to be biconvex or doubly convex.

22.1 Formal deﬁnition

Let G = (U ∪ V, E) be a bipartite graph, i.e, the vertex set is U ∪ V where U ∩ V = ∅. Let NG(v) denote the neighborhood of a vertex v ∈ V. The graph G is convex over U if and only if there exists a bijective mapping, f: U → {1, …, |U|}, such that for all v ∈ V, for any two vertices x,y ∈ NG(v) ⊆ U there does not exist a z ∉ NG(v) such that f(x) < f(z) < f(y).

22.2 See also

• Convex plane graph

22.3 References

• W. Lipski Jr.; Franco P. Preparata (August 1981). “Eﬃcient algorithms for ﬁnding maximum matchings in convex bipartite graphs and related problems”. Acta Informatica 15 (4): 329–346. doi:10.1007/BF00264533. Retrieved 2009-07-20.

• Ten-hwang Lai; Shu-shang Wei (April 1997). “Bipartite permutation graphs with application to the minimum buﬀer size problem”. Discrete Applied Mathematics 74 (1): 33–55. doi:10.1016/S0166-218X(96)00014-5. Retrieved 2009-07-20. • Jeremy P. Spinrad (2003). Eﬃcient graph representations. AMS Bookstore. p. 128. ISBN 978-0-8218-2815- 1. Retrieved 2009-07-20. • Andreas Brandstädt; Van Bang Le; Jeremy P. Spinrad (1999). Graph classes: a survey. SIAM. p. 94. ISBN 978-0-89871-432-6. Retrieved 2009-07-20.

92 Chapter 23

Critical graph

Not to be confused with the Critical path method, a concept in project management.

On the left-top a vertex critical graph with chromatic number 6; next all the N-1 subgraphs with chromatic number 5.

93 94 CHAPTER 23. CRITICAL GRAPH

In general the notion of criticality can refer to any measure. But in graph theory, when the term is used without any qualiﬁcation, it almost always refers to the chromatic number of a graph. Critical graphs are interesting because they are the minimal members in terms of chromatic number, which is a very important measure in graph theory. More precise deﬁnitions follow. A vertex or an edge is a critical element of a graph G if its deletion would decrease the chromatic number of G. Obviously such decrement can be no more than 1 in a graph. A critical graph is a graph in which every vertex or edge is a critical element. A k-critical graph is a critical graph with chromatic number k; a graph G with chromatic number k is k-vertex-critical if each of its vertices is a critical element. Some properties of a k-critical graph G with n vertices and m edges:

• G has only one component. • G is ﬁnite (this is the de Bruijn–Erdős theorem of de Bruijn & Erdős 1951). • δ(G) ≥ k − 1, that is, every vertex is adjacent to at least k − 1 others. More strongly, G is (k − 1)-edge-connected. See Lovász (1992) • If G is (k − 1)-regular, meaning every vertex is adjacent to exactly k − 1 others, then G is either Kk or an odd cycle . This is Brooks’ theorem; see Brooks & Tutte (1941)). • 2 m ≥ (k − 1) n + k − 3 (Dirac 1957). • 2 m ≥ (k − 1) n + [(k − 3)/(k2 − 3)] n (Gallai 1963a). • Either G may be decomposed into two smaller critical graphs, with an edge between every pair of vertices that includes one vertex from each of the two subgraphs, or G has at least 2k - 1 vertices (Gallai 1963b). More strongly, either G has a decomposition of this type, or for every vertex v of G there is a k-coloring in which v is the only vertex of its color and every other color class has at least two vertices (Stehlík 2003).

It is fairly easy to see that a graph G is vertex-critical if and only if for every vertex v, there is an optimal proper coloring in which v is a singleton color class. As Hajós (1961) showed, every k-critical graph may be formed from a complete graph Kk by combining the Hajós construction with an operation of identifying two nonadjacent vertices. The graphs formed in this way always require k colors in any proper coloring. A double-critical graph is a connected graph in which the deletion of any pair of adjacent vertices decreases the chromatic number by two. One open problem is to determine whether Kk is the only double-critical k-chromatic graph (Erdős 1966).

23.1 See also

• Factor-critical graph

23.2 References

• Brooks, R. L.; Tutte, W. T. (1941), “On colouring the nodes of a network”, Proceedings of the Cambridge Philosophical Society 37 (02): 194–197, doi:10.1017/S030500410002168X. • de Bruijn, N. G.; Erdős, P. (1951), “A colour problem for inﬁnite graphs and a problem in the theory of relations”, Nederl. Akad. Wetensch. Proc. Ser. A 54: 371–373. (Indag. Math. 13.) • Dirac, G. A. (1957), “A theorem of R. L. Brooks and a conjecture of H. Hadwiger”, Proceedings of the London Mathematical Society 7 (1): 161–195, doi:10.1112/plms/s3-7.1.161. • Gallai, T. (1963a), “Kritische Graphen I”, Publ. Math. Inst. Hungar. Acad. Sci. 8: 165–192. • Gallai, T. (1963b), “Kritische Graphen II”, Publ. Math. Inst. Hungar. Acad. Sci. 8: 373–395. 23.2. REFERENCES 95

• Hajós, G. (1961), "Über eine Konstruktion nicht n-färbbarer Graphen”, Wiss. Z. Martin-Luther-Univ. Halle- Wittenberg Math.-Natur. Reihe 10: 116–117. • Jensen, T. R.; Toft, B. (1995), Graph coloring problems, New York: Wiley-Interscience, ISBN 0-471-02865-7.

• Stehlík, Matěj (2003), “Critical graphs with connected complements”, Journal of Combinatorial Theory, Series B 89 (2): 189–194, doi:10.1016/S0095-8956(03)00069-8, MR 2017723.

• Lovász, László (1992), “Solution to Exercise 9.21”, Combinatorial Problems and Exercises (Second Edition), North-Holland.

• Erdős, Paul (1966), “Problem 2”, In Theory of Graphs, Proc. Colloq., Tihany, p. 361. Chapter 24

Cubic graph

Not to be confused with graphs of cubic functions, hypercube graphs, cubical graph. In the mathematical ﬁeld of graph theory, a cubic graph is a graph in which all vertices have degree three. In other

The Petersen graph is a Cubic graph. words a cubic graph is a 3-regular graph. Cubic graphs are also called trivalent graphs.

96 24.1. SYMMETRY 97

The complete bipartite graph K3,3 is an example of a bicubic graph

A bicubic graph is a cubic bipartite graph.

24.1 Symmetry

In 1932, Ronald M. Foster began collecting examples of cubic symmetric graphs, forming the start of the Foster census.[1] Many well-known individual graphs are cubic and symmetric, including the utility graph, the Petersen graph, the Heawood graph, the Möbius–Kantor graph, the Pappus graph, the Desargues graph, the Nauru graph, the Coxeter graph, the Tutte–Coxeter graph, the Dyck graph, the Foster graph and the Biggs-Smith graph. W. T. Tutte classiﬁed the symmetric cubic graphs by the smallest integer number s such that each two oriented paths of length s can be mapped to each other by exactly one symmetry of the graph. He showed that s is at most 5, and provided examples of graphs with each possible value of s from 1 to 5.[2] Semi-symmetric cubic graphs include the Gray graph (the smallest semi-symmetric cubic graph), the Ljubljana graph, and the Tutte 12-cage. The Frucht graph is one of the two smallest cubic graphs without any symmetries: it possesses only a single graph automorphism, the identity automorphism.

24.2 Coloring and independent sets

According to Brooks’ theorem every connected cubic graph other than the complete graph K4 can be colored with at most three colors. Therefore, every connected cubic graph other than K4 has an independent set of at least n/3 98 CHAPTER 24. CUBIC GRAPH vertices, where n is the number of vertices in the graph: for instance, the largest color class in a 3-coloring has at least this many vertices. According to Vizing’s theorem every cubic graph needs either three or four colors for an edge coloring. A 3-edge- coloring is known as a Tait coloring, and forms a partition of the edges of the graph into three perfect matchings. By König’s line coloring theorem every bicubic graph has a Tait coloring. The bridgeless cubic graphs that do not have a Tait coloring are known as snarks. They include the Petersen graph, Tietze’s graph, the Blanuša snarks, the ﬂower snark, the double-star snark, the Szekeres snark and the Watkins snark. There is an inﬁnite number of distinct snarks.[3]

24.3 Topology and geometry

Cubic graphs arise naturally in topology in several ways. For example, if one considers a graph to be a 1-dimensional CW complex, cubic graphs are generic in that most 1-cell attaching maps are disjoint from the 0-skeleton of the graph. Cubic graphs are also formed as the graphs of simple polyhedra in three dimensions, polyhedra such as the regular dodecahedron with the property that three faces meet at every vertex. An arbitrary graph embedding on a two-dimensional surface may be represented as a cubic graph structure known as a graph-encoded map. In this structure, each vertex of a cubic graph represents a flag of the embedding, a mutually incident triple of a vertex, edge, and face of the surface. The three neighbors of each flag are the three flags that may be obtained from it by changing one of the members of this mutually incident triple and leaving the other two members unchanged.[4]

24.4 Hamiltonicity

There has been much research on Hamiltonicity of cubic graphs. In 1880, P.G. Tait conjectured that every cubic polyhedral graph has a Hamiltonian circuit. William Thomas Tutte provided a counter-example to Tait’s conjecture, the 46-vertex Tutte graph, in 1946. In 1971, Tutte conjectured that all bicubic graphs are Hamiltonian. However, Joseph Horton provided a counterexample on 96 vertices, the Horton graph.[5] Later, Mark Ellingham constructed two more counterexamples: the Ellingham-Horton graphs.[6][7] Barnette’s conjecture, a still-open combination of Tait’s and Tutte’s conjecture, states that every bicubic polyhedral graph is Hamiltonian. When a cubic graph is Hamiltonian, LCF notation allows it to be represented concisely. If a cubic graph is chosen uniformly at random among all n-vertex cubic graphs, then it is very likely to be Hamiltonian: the proportion of the n-vertex cubic graphs that are Hamiltonian tends to one in the limit as n goes to inﬁnity.[8] David Eppstein conjectured that every n-vertex cubic graph has at most 2n/3 (approximately 1.260n) distinct Hamilto- nian cycles, and provided examples of cubic graphs with that many cycles.[9] The best proven estimate for the number of distinct Hamiltonian cycles is O(1.276n) .[10]

24.5 Other properties

The pathwidth of any n-vertex cubic graph is at most n/6. However, the best known lower bound on the pathwidth of cubic graphs is smaller, 0.082n.[11] It follows from the handshaking lemma, proven by Leonhard Euler in 1736 as part of the ﬁrst paper on graph theory, that every cubic graph has an even number of vertices. Petersen’s theorem states that every cubic bridgeless graph has a perfect matching.[12] Lovász and Plummer conjectured that every cubic bridgeless graph has an exponential number of perfect matchings. The conjecture was recently proved, showing that every cubic bridgeless graph with n vertices has at least 2n/3656 perfect matchings.[13] 24.6. ALGORITHMS AND COMPLEXITY 99

24.6 Algorithms and complexity

Several researchers have studied the complexity of exponential time algorithms restricted to cubic graphs. For instance, by applying dynamic programming to a path decomposition of the graph, Fomin and Høie showed how to ﬁnd their maximum independent sets in time O(2n/6 + o(n)).[11] The travelling salesman problem in cubic graphs can be solved in time O(1.2312n) and polynomial space.[14][15] Several important graph optimization problems are APX hard, meaning that, although they have approximation algorithms whose approximation ratio is bounded by a constant, they do not have polynomial time approximation schemes whose approximation ratio tends to 1 unless P=NP. These include the problems of ﬁnding a minimum vertex cover, maximum independent set, minimum dominating set, and maximum cut.[16] The crossing number (the minimum number of edges which cross in any graph drawing) of a cubic graph is also NP-hard for cubic graphs but may be approximated.[17] The Travelling Salesman Problem on cubic graphs has been proven to be NP-hard to approximate to within any factor less than 1153/1152.[18]

24.7 See also

• Table of simple cubic graphs

24.8 References

[1] Foster, R. M. (1932), “Geometrical Circuits of Electrical Networks”, Transactions of the American Institute of Electrical Engineers 51 (2): 309–317, doi:10.1109/T-AIEE.1932.5056068.

[2] Tutte, W. T. (1959), “On the symmetry of cubic graphs”, Canad. J. Math 11: 621–624, doi:10.4153/CJM-1959-057-2.

[3] Isaacs, R. (1975), “Inﬁnite families of nontrivial trivalent graphs which are not Tait colorable”, American Mathematical Monthly 82 (3): 221–239, doi:10.2307/2319844, JSTOR 2319844.

[4] Bonnington, C. Paul; Little, Charles H. C. (1995), The Foundations of Topological Graph Theory, Springer-Verlag.

[5] Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York: North Holland, p. 240, 1976.

[6] Ellingham, M. N. “Non-Hamiltonian 3-Connected Cubic Partite Graphs."Research Report No. 28, Dept. of Math., Univ. Melbourne, Melbourne, 1981.

[7] Ellingham, M. N.; Horton, J. D. (1983), “Non-Hamiltonian 3-connected cubic bipartite graphs”, Journal of Combinatorial Theory, Series B 34 (3): 350–353, doi:10.1016/0095-8956(83)90046-1.

[8] Robinson, R.W.; Wormald, N.C. (1994), “Almost all regular graphs are Hamiltonian”, Random Structures and Algorithms 5 (2): 363–374, doi:10.1002/rsa.3240050209.

[9] Eppstein, David (2007), “The traveling salesman problem for cubic graphs” (PDF), Journal of Graph Algorithms and Applications 11 (1): 61–81, arXiv:cs.DS/0302030, doi:10.7155/jgaa.00137.

[10] Gebauer, H. (2008), “On the number of Hamilton cycles in bounded degree graphs”, Proc. 4th Workshop on Analytic Algorithmics and Combinatorics (ANALCO '08) (PDF).

[11] Fomin, Fedor V.; Høie, Kjartan (2006), “Pathwidth of cubic graphs and exact algorithms”, Information Processing Letters 97 (5): 191–196, doi:10.1016/j.ipl.2005.10.012.

[12] Petersen, Julius Peter Christian (1891), “Die Theorie der regulären Graphs (The theory of regular graphs)", Acta Mathe- matica 15 (15): 193–220, doi:10.1007/BF02392606.

[13] Esperet, Louis; Kardoš, František; King, Andrew D.; Kráľ, Daniel; Norine, Serguei (2011), “Exponentially many perfect matchings in cubic graphs”, Advances in Mathematics 227 (4): 1646–1664, doi:10.1016/j.aim.2011.03.015.

[14] Xiao, Mingyu; Nagamochi, Hiroshi (2013), “An Exact Algorithm for TSP in Degree-3 Graphs via Circuit Procedure and Amortization on Connectivity Structure”, Theory and Applications of Models of Computation, Lecture Notes in Computer Science 7876, Springer-Verlag, pp. 96–107, doi:10.1007/978-3-642-38236-9_10, ISBN 978-3-642-38236-9.

[15] Xiao, Mingyu; Nagamochi, Hiroshi (2012), An Exact Algorithm for TSP in Degree-3 Graphs via Circuit Procedure and Amortization on Connectivity Structure, arXiv:1212.6831. 100 CHAPTER 24. CUBIC GRAPH

[16] Alimonti, Paola; Kann, Viggo (2000), “Some APX-completeness results for cubic graphs”, Theoretical Computer Science 237 (1–2): 123–134, doi:10.1016/S0304-3975(98)00158-3.

[17] Hliněný, Petr (2006), “Crossing number is hard for cubic graphs”, Journal of Combinatorial Theory, Series B 96 (4): 455–471, doi:10.1016/j.jctb.2005.09.009.

[18] Karpinski, Marek; Schmied, Richard (2013), Approximation Hardness of Graphic TSP on Cubic Graphs, arXiv:1304.6800.

24.9 External links

• Royle, Gordon. “Cubic symmetric graphs (The Foster Census)". • Weisstein, Eric W., “Bicubic Graph”, MathWorld.

• Weisstein, Eric W., “Cubic Graph”, MathWorld. Chapter 25

Dense graph

In mathematics, a dense graph is a graph in which the number of edges is close to the maximal number of edges. The opposite, a graph with only a few edges, is a sparse graph. The distinction between sparse and dense graphs is rather vague, and depends on the context. For undirected simple graphs, the graph density is deﬁned as:

2|E| D = |V | (|V | − 1)

For directed simple graphs, the graph density is deﬁned as:

|E| D = |V | (|V | − 1) where E is the number of edges and V is the number of vertices in the graph. The maximum number of edges is ½ |V| (|V|−1), so the maximal density is 1 (for complete graphs) and the minimal density is 0 (Coleman & Moré 1983).

25.1 Upper density

Upper density is an extension of the concept of graph density defined above from finite graphs to infinite graphs. Intuitively, an infinite graph has arbitrarily large finite subgraphs with any density less than its upper density, and does not have arbitrarily large finite subgraphs with density greater than its upper density. Formally, the upper density of a graph G is the infimum of the values α such that the finite subgraphs of G with density α have a bounded number of vertices. It can be shown using the Erdős–Stone theorem that the upper density can only be 1 or one of the superparticular ratios 0, 1/2, 2/3, 3/4, 4/5, ... n/(n + 1), ... (see, e.g., Diestel, p. 189).

25.2 Sparse and tight graphs

Lee & Streinu (2008) and Streinu & Theran (2009) deﬁne a graph as being (k,l)-sparse if every nonempty subgraph with n vertices has at most kn − l edges, and (k,l)-tight if it is (k,l)-sparse and has exactly kn − l edges. Thus trees are exactly the (1,1)-tight graphs, forests are exactly the (1,1)-sparse graphs, and graphs with arboricity k are exactly the (k,k)-sparse graphs. Pseudoforests are exactly the (1,0)-sparse graphs, and the Laman graphs arising in rigidity theory are exactly the (2,3)-tight graphs. Other graph families not characterized by their sparsity can also be described in this way. For instance the facts that any planar graph with n vertices has at most 3n - 6 edges, and that any subgraph of a planar graph is planar, together imply that the planar graphs are (3,6)-sparse. However, not every (3,6)-sparse graph is planar. Similarly, outerplanar graphs are (2,3)-sparse and planar bipartite graphs are (2,4)-sparse.

101 102 CHAPTER 25. DENSE GRAPH

Streinu and Theran show that testing (k,l)-sparsity may be performed in polynomial time when k and l are integers and 0 ≤ l < 2k. For a graph family, the existence of k and l such that the graphs in the family are all (k,l)-sparse is equivalent to the graphs in the family having bounded degeneracy or having bounded arboricity. More precisely, it follows from a result of Nash-Williams (1964) that the graphs of arboricity at most a are exactly the (a,a)-sparse graphs. Similarly, the graphs of degeneracy at most d are exactly the ((d + 1)/2,1)-sparse graphs.

25.3 Sparse and dense classes of graphs

Nešetřil & Ossona de Mendez (2010) considered that the sparsity/density dichotomy makes it necessary to consider inﬁnite graph classes instead of single graph instances. They deﬁned somewhere dense graph classes as those classes of graphs for which there exists a threshold t such that every complete graph appears as a t-subdivision in a subgraph of a graph in the class. To the contrary, if such a threshold does not exist, the class is nowhere dense. Properties of the nowhere dense vs somewhere dense dichotomy are discussed in Nešetřil & Ossona de Mendez (2012). The classes of graphs with bounded degeneracy and of nowhere dense graphs are both included in the biclique-free graphs, graph families that exclude some complete bipartite graph as a subgraph (Telle & Villanger 2012).

25.4 See also

• Bounded expansion • Dense subgraph

25.5 References

• Paul E. Black, Sparse graph, from Dictionary of Algorithms and Data Structures, Paul E. Black (ed.), NIST. Retrieved on 29 September 2005. • Coleman, Thomas F.; Moré, Jorge J. (1983), “Estimation of sparse Jacobian matrices and graph coloring Problems”, SIAM Journal on Numerical Analysis 20 (1): 187–209, doi:10.1137/0720013. • Diestel, Reinhard (2005), Graph Theory, Graduate Texts in Mathematics, Springer-Verlag, ISBN 3-540- 26183-4, OCLC 181535575. • Lee, Audrey; Streinu, Ileana (2008), “Pebble game algorithms and sparse graphs”, Discrete Mathematics 308 (8): 1425–1437, doi:10.1016/j.disc.2007.07.104, MR 2392060. • Nash-Williams, C. St. J. A. (1964), “Decomposition of ﬁnite graphs into forests”, Journal of the London Mathematical Society 39 (1): 12, doi:10.1112/jlms/s1-39.1.12, MR 0161333 • Preiss, ﬁrst (1998), Data Structures and Algorithms with Object-Oriented Design Patterns in C++, John Wiley & Sons, ISBN 0-471-24134-2. • Nešetřil, Jaroslav; Ossona de Mendez, Patrice (2010), From Sparse Graphs to Nowhere Dense Structures: De- compositions, Independence, Dualities and Limits, European Mathematical Society, pp. 135–165 • Nešetřil, Jaroslav; Ossona de Mendez, Patrice (2012), Sparsity: Graphs, Structures, and Algorithms, Algorithms and Combinatorics 28, Heidelberg: Springer, doi:10.1007/978-3-642-27875-4, ISBN 978-3-642-27874-7, MR 2920058. • Streinu, I.; Theran, L. (2009), “Sparse hypergraphs and pebble game algorithms”, European Journal of Com- binatorics 30 (8): 1944–1964, arXiv:math/0703921, doi:10.1016/j.ejc.2008.12.018. • Telle, Jan Arne; Villanger, Yngve (2012), “FPT algorithms for domination in biclique-free graphs”, in Ep- stein, Leah; Ferragina, Paolo, Algorithms – ESA 2012: 20th Annual European Symposium, Ljubljana, Slove- nia, September 10–12, 2012, Proceedings, Lecture Notes in Computer Science 7501, Springer, pp. 802–812, doi:10.1007/978-3-642-33090-2_69. Chapter 26

Descartes snark

In the mathematical ﬁeld of graph theory, a Descartes snark is an undirected graph with 210 vertices and 315 edges. It is a snark, ﬁrst discovered by William Tutte in 1948 under the pseudonym Blanche Descartes.[1] A Descartes snark is obtained from the Petersen graph by replacing each vertex with a nonagon and each edge with a particular graph closely related to the Petersen graph. Because there are multiple ways to perform this procedure, there are multiple Descartes snarks.

26.1 Notes

[1] Descartes, Blanche. "Network Colorings,” The Mathematical Gazette (London, 32:299. p. 67–69, 1948.

103 Chapter 27

Disperser

A disperser is a one-sided extractor.[1] Where an extractor requires that every event gets the same probability under the uniform distribution and the extracted distribution, only the latter is required for a disperser. So for a disperser, ⊆ { }m − an event A 0, 1 we have: P rUm [A] > 1 ϵ Deﬁnition (Disperser): A (k, ϵ) -disperser is a function Dis : {0, 1}n × {0, 1}d → {0, 1}m n such that for every distribution X on {0, 1} with H∞(X) ≥ k the support of the distribution Dis(X,Ud) is of size at least (1 − ϵ)2m .

27.1 Graph theory

An (N, M, D, K, e)-disperser is a bipartite graph with N vertices on the left side, each with degree D, and M vertices on the right side, such that every subset of K vertices on the left side is connected to more than (1 − e)M vertices on the right. An extractor is a related type of graph that guarantees an even stronger property; every (N, M, D, K, e)-extractor is also an (N, M, D, K, e)-disperser.

27.2 Other meanings

A disperser is a high-speed mixing device used to disperse or dissolve pigments and other solids into a liquid.

27.3 See also

• Expander graph

27.4 References

[1] Ronen Shaltiel. Recent developments in explicit construction of extractors. P. 7.

104 Chapter 28

Distance-hereditary graph

A distance-hereditary graph.

In graph theory, a branch of discrete mathematics, a distance-hereditary graph (also called a completely separable graph)[1] is a graph in which the distances in any connected induced subgraph are the same as they are in the original graph. Thus, any induced subgraph inherits the distances of the larger graph. Distance-hereditary graphs were named and ﬁrst studied by Howorka (1977), although an equivalent class of graphs was already shown to be perfect in 1970 by Olaru and Sachs.[2] It has been known for some time that the distance-hereditary graphs constitute an intersection class of graphs,[3] but no intersection model was known until one was given by Gioan & Paul (2012).

105 106 CHAPTER 28. DISTANCE-HEREDITARY GRAPH

28.1 Deﬁnition and characterization

The original deﬁnition of a distance-hereditary graph is a graph G such that, if any two vertices u and v belong to a connected induced subgraph H of G, then some shortest path connecting u and v in G must be a subgraph of H, so that the distance between u and v in H is the same as the distance in G. Distance-hereditary graphs can also be characterized in several other equivalent ways:[4]

• They are the graphs in which every induced path is a shortest path, or equivalently the graphs in which every non-shortest path has at least one edge connecting two non-consecutive path vertices.

• They are the graphs in which every cycle of length at least ﬁve has two or more diagonals, and in which every cycle of length exactly ﬁve has at least one pair of crossing diagonals.

• They are the graphs in which every cycle of length ﬁve or more has at least one pair of crossing diagonals.

• They are the graphs in which, for every four vertices u, v, w, and x, at least two of the three sums of distances d(u,v)+d(w,x), d(u,w)+d(v,x), and d(u,x)+d(v,w) are equal to each other.

• They are the graphs that do not have as isometric subgraphs any cycle of length ﬁve or more, or any of three other graphs: a 5-cycle with one chord, a 5-cycle with two non-crossing chords, and a 6-cycle with a chord connecting opposite vertices.

• They are the graphs that can be built up from a single vertex by a sequence of the following three operations, as shown in the illustration: 1. Add a new pendant vertex connected by a single edge to an existing vertex of the graph. 2. Replace any vertex of the graph by a pair of vertices, each of which has the same set of neighbors as the replaced vertex. The new pair of vertices are called false twins of each other. 3. Replace any vertex of the graph by a pair of vertices, each of which has as its neighbors the neighbors of the replaced vertex together with the other vertex of the pair. The new pair of vertices are called true twins of each other.

• They are the graphs that can be completely decomposed into cliques and stars (complete bipartite graphs K₁,q) by a split decomposition. In this decomposition, one ﬁnds a partition of the graph into two subsets, such that the edges separating the two subsets form a complete bipartite subgraph, forms two smaller graphs by replacing each of the two sides of the partition by a single vertex, and recursively partitions these two subgraphs.[5]

• They are the graphs that have rank-width one, where the rank-width of a graph is deﬁned as the minimum, over all hierarchical partitions of the vertices of the graph, of the maximum rank among certain submatrices of the graph’s adjacency matrix determined by the partition.[6]

28.2 Relation to other graph families

Every distance-hereditary graph is a perfect graph[7] and more speciﬁcally a perfectly orderable graph.[8] Every distance-hereditary graph is also a parity graph, a graph in which every two induced paths between the same pair of vertices both have odd length or both have even length.[9] Every even power of a distance-hereditary graph G (that is, the graph G2i formed by connecting pairs of vertices at distance at most 2i in G) is a chordal graph.[10] Every distance-hereditary graph can be represented as the intersection graph of chords on a circle, forming a circle graph. This can be seen by building up the graph by adding pendant vertices, false twins, and true twins, at each step building up a corresponding set of chords representing the graph. Adding a pendant vertex corresponds to adding a chord near the endpoints of an existing chord so that it crosses only that chord; adding false twins corresponds to replacing a chord by two parallel chords crossing the same set of other chords; and adding true twins corresponds to replacing a chord by two chords that cross each other but are nearly parallel and cross the same set of other chords. 28.3. ALGORITHMS 107

pendant false true vertex twins twins

Three operations by which any distance-hereditary graph can be constructed.

A distance-hereditary graph is bipartite if and only if it is triangle-free. Bipartite distance-hereditary graphs can be built up from a single vertex by adding only pendant vertices and false twins, since any true twin would form a triangle, but the pendant vertex and false twin operations preserve bipartiteness. The graphs that can be built from a single vertex by pendant vertices and true twins, without any false twin operations, are the chordal distance-hereditary graphs, also called ptolemaic graphs; they include as a special case the block graphs. The graphs that can be built from a single vertex by false twin and true twin operations, without any pendant vertices, are the cographs, which are therefore distance-hereditary; the cographs are exactly the disjoint unions of diameter-2 distance-hereditary graphs. The neighborhood of any vertex in a distance-hereditary graph is a cograph. The transitive closure of the directed graph formed by choosing any set of orientations for the edges of any tree is distance-hereditary; the special case in which the tree is oriented consistently away from some vertex forms a subclass of distance-hereditary graphs known as the trivially perfect graphs, which are also called chordal cographs.[11]

28.3 Algorithms

Distance-hereditary graphs can be recognized, and parsed into a sequence of pendant vertex and twin operations, in linear time.[12] Because distance-hereditary graphs are perfect, some optimization problems can be solved in polynomial time for them despite being NP-hard for more general classes of graphs. Thus, it is possible in polynomial time to find the maximum clique or maximum independent set in a distance-hereditary graph, or to find an optimal graph coloring of any distance-hereditary graph.[13] Because distance-hereditary graphs are circle graphs, they inherit polynomial time 108 CHAPTER 28. DISTANCE-HEREDITARY GRAPH algorithms for circle graphs; for instance, it is possible determine in polynomial time the treewidth of any circle graph and therefore of any distance-hereditary graph.[14] Additionally, the clique-width of any distance-hereditary graph is at most three.[15] As a consequence, by Courcelle’s theorem, efficient dynamic programming algorithms exist for many problems on these graphs.[16] Several other optimization problems can also be solved more efficiently using algorithms specifically designed for distance-hereditary graphs. As D'Atri & Moscarini (1988) show, a minimum connected dominating set (or equivalently a spanning tree with the maximum possible number of leaves) can be found in polynomial time on a distance- hereditary graph. A Hamiltonian cycle or Hamiltonian path of any distance-hereditary graph can also be found in polynomial time.[17]

28.4 Notes

[1] Hammer & Maﬀray (1990).

[2] Olaru and Sachs showed the α-perfection of the graphs in which every cycle of length ﬁve or more has a pair of crossing diagonals (Sachs 1970, Theorem 5). By Lovász (1972), α-perfection is an equivalent form of deﬁnition of perfect graphs.

[3] McKee & McMorris (1999)

[4] Howorka (1977); Bandelt & Mulder (1986); Hammer & Maﬀray (1990); Brandstädt, Le & Spinrad (1999), Theorem 10.1.1, p. 147.

[5] Gioan & Paul (2012). A closely related decomposition was used for graph drawing by Eppstein, Goodrich & Meng (2006) and (for bipartite distance-hereditary graphs) by Hui, Schaefer & Štefankovič (2004).

[6] Oum (2005).

[7] Howorka (1977).

[8] Brandstädt, Le & Spinrad (1999), pp. 70–71 and 82.

[9] Brandstädt, Le & Spinrad (1999), p.45.

[10] Brandstädt, Le & Spinrad (1999), Theorem 10.6.14, p.164.

[11] Cornelsen & Di Stefano (2005).

[12] Damiand, Habib & Paul (2001); Gioan & Paul (2012). An earlier linear time bound was claimed by Hammer & Maﬀray (1990) but it was discovered to be erroneous by Damiand.

[13] Cogis & Thierry (2005) present a simple direct algorithm for maximum weighted independent sets in distance-hereditary graphs, based on parsing the graph into pendant vertices and twins, correcting a previous attempt at such an algorithm by Hammer & Maﬀray (1990). Because distance-hereditary graphs are perfectly orderable, they can be optimally colored in linear time by using LexBFS to ﬁnd a perfect ordering and then applying a greedy coloring algorithm.

[14] Kloks (1996); Brandstädt, Le & Spinrad (1999), p. 170.

[15] Golumbic & Rotics (2000).

[16] Courcelle, Makowski & Rotics (2000); Espelage, Gurski & Wanke (2001).

[17] Hsieh et al. (2002); Müller & Nicolai (1993).

28.5 References

• Bandelt, Hans-Jürgen; Mulder, Henry Martyn (1986), “Distance-hereditary graphs”, Journal of Combinatorial Theory, Series B 41 (2): 182–208, doi:10.1016/0095-8956(86)90043-2, MR 0859310.

• Brandstädt, Andreas; Le, Van Bang; Spinrad, Jeremy (1999), Graph Classes: A Survey, SIAM Monographs on Discrete Mathematics and Applications, ISBN 0-89871-432-X.

• Cogis, O.; Thierry, E. (2005), “Computing maximum stable sets for distance-hereditary graphs”, Discrete Optimization 2 (2): 185–188, doi:10.1016/j.disopt.2005.03.004, MR 2155518. 28.5. REFERENCES 109

• Cornelsen, Sabine; Di Stefano, Gabriele (2005), “Treelike comparability graphs: characterization, recognition, and applications”, Proc. 30th Int. Worksh. Graph-Theoretic Concepts in Computer Science (WG 2004), Lecture Notes in Computer Science 3353, Springer-Verlag, pp. 46–57, MR 2158633. • Courcelle, B.; Makowski, J. A.; Rotics, U. (2000), “Linear time solvable optimization problems on graphs on bounded clique width”, Theory of Computing Systems 33 (2): 125–150, doi:10.1007/s002249910009. • D'Atri, Alessandro; Moscarini, Marina (1988), “Distance-hereditary graphs, Steiner trees, and connected domination”, SIAM Journal on Computing 17 (3): 521–538, doi:10.1137/0217032, MR 0941943. • Damiand, Guillaume; Habib, Michel; Paul, Christophe (2001), “A simple paradigm for graph recognition: application to cographs and distance hereditary graphs”, Theoretical Computer Science 263 (1–2): 99–111, doi:10.1016/S0304-3975(00)00234-6, MR 1846920. • Eppstein, David; Goodrich, Michael T.; Meng, Jeremy Yu (2006), “Delta-confluent drawings”, in Healy, Patrick; Nikolov, Nikola S., Proc. 13th Int. Symp. Graph Drawing (GD 2005), Lecture Notes in Computer Science 3843, Springer-Verlag, pp. 165–176, arXiv:cs.CG/0510024, MR 2244510. • Espelage, W.; Gurski, F.; Wanke, E. (2001), “How to solve NP-hard graph problems on clique-width bounded graphs in polynomial time”, Proc. 27th Int. Worksh. Graph-Theoretic Concepts in Computer Science (WG 2001), Lecture Notes in Computer Science 2204, Springer-Verlag, pp. 117–128. • Gioan, Emeric; Paul, Christophe (2012), “Split decomposition and graph-labelled trees: Characterizations and fully dynamic algorithms for totally decomposable graphs”, Discrete Applied Mathematics 160 (6): 708–733, arXiv:0810.1823, doi:10.1016/j.dam.2011.05.007. • Golumbic, Martin Charles; Rotics, Udi (2000), “On the clique-width of some perfect graph classes”, Interna- tional Journal of Foundations of Computer Science 11 (3): 423–443, doi:10.1142/S0129054100000260, MR 1792124. • Hammer, Peter L.; Maffray, Frédéric (1990), “Completely separable graphs”, Discrete Applied Mathematics 27 (1–2): 85–99, doi:10.1016/0166-218X(90)90131-U, MR 1055593. • Howorka, Edward (1977), “A characterization of distance-hereditary graphs”, The Quarterly Journal of Math- ematics. Oxford. Second Series 28 (112): 417–420, doi:10.1093/qmath/28.4.417, MR 0485544. • Hsieh, Sun-yuan; Ho, Chin-wen; Hsu, Tsan-sheng; Ko, Ming-tat (2002), “Efficient algorithms for the Hamilto- nian problem on distance-hereditary graphs”, Computing and Combinatorics: 8th Annual International Confer- ence, COCOON 2002 Singapore, August 15–17, 2002, Proceedings, Lecture Notes in Computer Science 2387, Springer-Verlag, pp. 51–75, doi:10.1007/3-540-45655-4_10, ISBN 978-3-540-43996-7, MR 2064504. • Hui, Peter; Schaefer, Marcus; Štefankovič, Daniel (2004), “Train tracks and confluent drawings”, in Pach, János, Proc. 12th Int. Symp. Graph Drawing (GD 2004), Lecture Notes in Computer Science 3383, Springer- Verlag, pp. 318–328. • Kloks, T. (1996), “Treewidth of circle graphs”, International Journal of Foundations of Computer Science 7 (2): 111–120, doi:10.1142/S0129054196000099. • Lovász, László (1972), “Normal hypergraphs and the perfect graph conjecture”, Discrete Mathematics 2 (3): 253–267, doi:10.1016/0012-365X(72)90006-4, MR 0302480. • McKee, Terry A.; McMorris, F. R. (1999), Topics in Intersection Graph Theory, SIAM Monographs on Dis- crete Mathematics and Applications 2, Philadelphia: Society for Industrial and Applied Mathematics, ISBN 0-89871-430-3, MR 1672910 • Müller, Haiko; Nicolai, Falk (1993), “Polynomial time algorithms for Hamiltonian problems on bipartite distance-hereditary graphs”, Information Processing Letters 46 (5): 225–230, doi:10.1016/0020-0190(93)90100- N, MR 1228792. • Oum, Sang-il (2005), “Rank-width and vertex-minors”, Journal of Combinatorial Theory, Series B 95 (1): 79–100, doi:10.1016/j.jctb.2005.03.003, MR 2156341. • Sachs, Horst (1970), “On the Berge conjecture concerning perfect graphs”, Combinatorial Structures and their Applications (Proc. Calgary Internat. Conf., Calgary, Alta., 1969), Gordon and Breach, pp. 377–384, MR 0272668. 110 CHAPTER 28. DISTANCE-HEREDITARY GRAPH

28.6 External links

• “Distance-hereditary graphs”, Information System on Graph Classes and their Inclusions]. Chapter 29

Distance-regular graph

In mathematics, a distance-regular graph is a regular graph such that for any two vertices v and w, the number of vertices at distance j from v and at distance k from w depends only upon j, k, and i = d(v, w). By considering the special case k = 1, one sees that in a distance-regular graph, for any two vertices v and w at distance i, the number of neighbors of w that are at distance j from v is the same. Conversely, it turns out that this special case implies the full deﬁnition of distance-regularity.[1] Therefore, an equivalent deﬁnition is that a distance-regular graph is a regular graph for which there exist integers bᵢ,cᵢ,i=0,...,d such that for any two vertices x,y with y in Gᵢ(x), there are exactly bᵢ neighbors of y in Gᵢ-₁(x) and cᵢ neighbors of y in Gᵢ₊₁(x), where Gᵢ(x) is the set of vertices y of G with d(x,y)=i (Brouwer et al., p. 434). The array of integers characterizing a distance-regular graph is known as its intersection array. Every distance-transitive graph is distance regular. Indeed, distance-regular graphs were introduced as a combinatorial generalization of distance-transitive graphs, having the numerical regularity properties of the latter without necessarily having a large automorphism group. A distance-regular graph with diameter 2 is strongly regular, and conversely (unless the graph is disconnected).

29.1 Intersection numbers

It is usual to use the following notation for a distance-regular graph G. The number of vertices is n. The number of neighbors of w (that is, vertices adjacent to w) whose distance from v is i, i + 1, and i − 1 is denoted by ai, bi, and ci, respectively; these are the intersection numbers of G. Obviously, a0 = 0, c0 = 0, and b0 equals k, the degree of any vertex. If G has ﬁnite diameter, then d denotes the diameter and we have bd = 0. Also we have that ai+bi+ci= k The numbers ai, bi, and ci are often displayed in a three-line array

  − c1 ··· cd−1 cd  ··· a0 a1 ad−1 ad , b0 b1 ··· bd−1 −

called the intersection array of G. They may also be formed into a tridiagonal matrix

  a0 b0 0 ··· 0 0   c1 a1 b1 ··· 0 0     0 c2 a2 ··· 0 0  B :=  ,  . . . . .   . . . . .    0 0 0 ··· ad−1 bd−1 0 0 0 ··· cd ad

called the intersection matrix.

111 112 CHAPTER 29. DISTANCE-REGULAR GRAPH

29.2 Distance adjacency matrices

Suppose G is a connected distance-regular graph. For each distance i = 1, ..., d, we can form a graph Gi in which vertices are adjacent if their distance in G equals i. Let Ai be the adjacency matrix of Gi. For instance, A1 is the adjacency matrix A of G. Also, let A0 = I, the identity matrix. This gives us d + 1 matrices A0, A1, ..., Ad, called the distance matrices of G. Their sum is the matrix J in which every entry is 1. There is an important product formula:

AAi = aiAi + biAi+1 + ciAi−1. From this formula it follows that each Ai is a polynomial function of A, of degree i, and that A satisﬁes a polynomial of degree d + 1. Furthermore, A has exactly d + 1 distinct eigenvalues, namely the eigenvalues of the intersection matrix B,of which the largest is k, the degree. The distance matrices span a vector subspace of the vector space of all n × n real matrices. It is a remarkable fact that the product Ai Aj of any two distance matrices is a linear combination of the distance matrices:

∑d k AiAj = pijAk. k=0 This means that the distance matrices generate an association scheme. The theory of association schemes is central to the study of distance-regular graphs. For instance, the fact that Ai is a polynomial function of A is a fact about association schemes.

29.3 Examples

• Complete graphs are distance regular with diameter 1 and degree v−1. • Cycles Cn are distance regular with k = 2, for all n > 2. • All Moore graphs, in particular the Petersen graph and the Hoﬀman-Singleton graph, are distance regular. • The Wells graph and Sylvester graph. • Strongly regular graphs are distance regular. • The odd graphs are distance regular. • The collinearity graph of every regular near polygon is distance-regular.

29.3.1 Cubic distance-regular graphs

There are 13 distance-regular cubic graphs: K4 (or tetrahedron), K₃,₃, the Petersen graph, the cube, the Heawood graph, the Pappus graph, the Coxeter graph, the Tutte–Coxeter graph, the dodecahedron, the Desargues graph, Tutte 12-cage, the Biggs–Smith graph, and the Foster graph.

29.4 Notes

[1] A.E. Brouwer, A.M. Cohen, and A. Neumaier (1989), Distance Regular Graphs. Berlin, New York: Springer-Verlag. ISBN 3-540-50619-5, ISBN 0-387-50619-5

29.5 Further reading

• Godsil, C. D. (1993). Algebraic combinatorics. Chapman and Hall Mathematics Series. New York: Chapman and Hall. pp. xvi+362. ISBN 0-412-04131-6. MR 1220704. Chapter 30

Distance-transitive graph

The Biggs-Smith graph, the largest 3-regular distance-transitive graph.

In the mathematical ﬁeld of graph theory, a distance-transitive graph is a graph such that, given any two vertices v and w at any distance i, and any other two vertices x and y at the same distance, there is an automorphism of the graph that carries v to x and w to y. A distance transitive graph is vertex transitive and symmetric as well as distance regular. A distance-transitive graph is interesting partly because it has a large automorphism group. Some interesting ﬁnite

113 114 CHAPTER 30. DISTANCE-TRANSITIVE GRAPH groups are the automorphism groups of distance-transitive graphs, especially of those whose diameter is 2. Distance-transitive graphs were first defined in 1971 by Norman L. Biggs and D. H. Smith, who showed that there are only 12 finite trivalent distance-transitive graphs. These are: Independently in 1969 a Russian group led by Georgy Adelson-Velsky showed that there exist graphs that are distance- regular but not distance-transitive. The only graph of this type with degree three is the 126-vertex Tutte 12-cage. The smallest distance-regular graph that is not distance-transitive is the Shrikhande graph. Complete lists of distance- transitive graphs are known for some degrees larger than three, but the classification of distance-transitive graphs with arbitrarily large vertex degree remains open. The simplest asymptotic family of examples of distance-transitive graphs is the Hypercube graphs. Other families are the folded cube graphs and the square rook’s graphs. All three of these families have arbitrarily high degree.

30.1 References

Early works

• Adel’son-Vel’skii, G. M.; Veĭsfeĭler, B. Ju.; Leman, A. A.; Faradžev, I. A. (1969), “An example of a graph which has no transitive group of automorphisms”, Doklady Akademii Nauk SSSR 185: 975–976, MR 0244107.

• Biggs, Norman (1971), “Intersection matrices for linear graphs”, Combinatorial Mathematics and its Applica- tions (Proc. Conf., Oxford, 1969), London: Academic Press, pp. 15–23, MR 0285421.

• Biggs, Norman (1971), Finite Groups of Automorphisms, London Mathematical Society Lecture Note Series 6, London & New York: Cambridge University Press, MR 0327563. • Biggs, N. L.; Smith, D. H. (1971), “On trivalent graphs”, Bulletin of the London Mathematical Society 3 (2): 155–158, doi:10.1112/blms/3.2.155, MR 0286693. • Smith, D. H. (1971), “Primitive and imprimitive graphs”, The Quarterly Journal of Mathematics. Oxford. Second Series 22 (4): 551–557, doi:10.1093/qmath/22.4.551, MR 0327584.

Surveys

• Biggs, N. L. (1993), “Distance-Transitive Graphs”, Algebraic Graph Theory (2nd ed.), Cambridge University Press, pp. 155–163, chapter 20. • Van Bon, John (2007), “Finite primitive distance-transitive graphs”, European Journal of Combinatorics 28 (2): 517–532, doi:10.1016/j.ejc.2005.04.014, MR 2287450.

• Brouwer, A. E.; Cohen, A. M.; Neumaier, A. (1989), “Distance-Transitive Graphs”, Distance-Regular Graphs, New York: Springer-Verlag, pp. 214–234, chapter 7.

• Cohen, A. M. Cohen (2004), “Distance-transitive graphs”, in Beineke, L. W.; Wilson, R. J., Topics in Algebraic Graph Theory, Encyclopedia of Mathematics and its Applications 102, Cambridge University Press, pp. 222– 249. • Godsil, C.; Royle, G. (2001), “Distance-Transitive Graphs”, Algebraic Graph Theory, New York: Springer- Verlag, pp. 66–69, section 4.5. • Ivanov, A. A. (1992), “Distance-transitive graphs and their classiﬁcation”, in Faradžev, I. A.; Ivanov, A. A.; Klin, M.; et al., The Algebraic Theory of Combinatorial Objects, Math. Appl. (Soviet Series) 84, Dordrecht: Kluwer, pp. 283–378, MR 1321634.

30.2 External links

• Weisstein, Eric W., “Distance-Transitive Graph”, MathWorld. Chapter 31

Domination perfect graph

For Dominator in control flow graphs, see Dominator (graph theory). In graph theory, a dominating set for a graph G = (V, E) is a subset D of V such that every vertex not in D is adjacent to at least one member of D. The domination number γ(G) is the number of vertices in a smallest dominating set for G. The dominating set problem concerns testing whether γ(G) ≤ K for a given graph G and input K; it is a classical NP-complete decision problem in computational complexity theory (Garey & Johnson 1979). Therefore it is believed that there is no efficient algorithm that finds a smallest dominating set for a given graph. Figures (a)–(c) on the right show three examples of dominating sets for a graph. In each example, each white vertex is adjacent to at least one red vertex, and it is said that the white vertex is dominated by the red vertex. The domination number of this graph is 2: the examples (b) and (c) show that there is a dominating set with 2 vertices, and it can be checked that there is no dominating set with only 1 vertex for this graph.

31.1 History

As Hedetniemi & Laskar (1990) note, the domination problem was studied from the 1950s onwards, but the rate of research on domination signiﬁcantly increased in the mid-1970s. Their bibliography lists over 300 papers related to domination in graphs.

31.2 Bounds

Let G be a graph with n ≥ 1 vertices and let Δ be the maximum degree of the graph. The following bounds on γ(G) are known (Haynes, Hedetniemi & Slater 1998a, Chapter 2):

• One vertex can dominate at most Δ other vertices; therefore γ(G) ≥ n/(1 + Δ).

• The set of all vertices is a dominating set in any graph; therefore γ(G) ≤ n.

• If there are no isolated vertices in G, then there are two disjoint dominating sets in G; see domatic partition for details. Therefore in any graph without isolated vertices it holds that γ(G) ≤ n/2.

31.3 Independent domination

Dominating sets are closely related to independent sets: an independent set is also a dominating set if and only if it is a maximal independent set, so any maximal independent set in a graph is necessarily also a minimal dominating set. Thus, the smallest maximal independent set is greater in size than the smallest independent dominating set. The independent domination number i(G) of a graph G is the size of the smallest independent dominating set (or, equivalently, the size of the smallest maximal independent set).

115 116 CHAPTER 31. DOMINATION PERFECT GRAPH

(a)

(b)

(c)

Dominating sets (red vertices).

The minimum dominating set in a graph will not necessarily be independent, but the size of a minimum dominating set is always less than or equal to the size of a minimum maximal independent set, that is, γ(G) ≤ i(G). There are graph families in which a minimum maximal independent set is a minimum dominating set. For example, Allan & Laskar (1978) show that γ(G) = i(G) if G is a claw-free graph. A graph G is called a domination-perfect graph if γ(H) = i(H) in every induced subgraph H of G. Since an induced subgraph of a claw-free graph is claw-free, it follows that every claw-free graphs is also domination-perfect (Faudree, 31.4. ALGORITHMS AND COMPUTATIONAL COMPLEXITY 117

Flandrin & Ryjáček 1997).

31.3.1 Examples

Figures (a) and (b) are independent dominating sets, while ﬁgure (c) illustrates a dominating set that is not an independent set. For any graph G, its line graph L(G) is claw-free, and hence a minimum maximal independent set in L(G) is also a minimum dominating set in L(G). An independent set in L(G) corresponds to a matching in G, and a dominating set in L(G) corresponds to an edge dominating set in G. Therefore a minimum maximal matching has the same size as a minimum edge dominating set.

31.4 Algorithms and computational complexity

There exists a pair of polynomial-time L-reductions between the minimum dominating set problem and the set cover problem (Kann 1992, pp. 108–109). These reductions (see below) show that an efficient algorithm for the minimum dominating set problem would provide an efficient algorithm for the set cover problem and vice versa. Moreover, the reductions preserve the approximation ratio: for any α, a polynomial-time α-approximation algorithm for minimum dominating sets would provide a polynomial-time α-approximation algorithm for the set cover problem and vice versa. Both problems are in fact Log-APX-complete.[1] The set cover problem is a well-known NP-hard problem – the decision version of set covering was one of Karp’s 21 NP-complete problems, which were shown to be NP-complete already in 1972. Hence the reductions show that the dominating set problem is NP-hard as well. The approximability of set covering is also well understood: a logarithmic approximation factor can be found by using a simple greedy algorithm, and finding a sublogarithmic approximation factor is NP-hard. More specifically, the greedy algorithm provides a factor 1 + log |V| approximation of a minimum dominating set, and Raz & Safra (1997) show that no algorithm can achieve an approximation factor better than c log |V| for some c > 0 unless P = NP.

31.4.1 L-reductions

The following pair of reductions (Kann 1992, pp. 108–109) shows that the minimum dominating set problem and the set cover problem are equivalent under L-reductions: given an instance of one problem, we can construct an equivalent instance of the other problem. From dominating set to set covering. Given a graph G = (V, E) with V = {1, 2, ..., n}, construct a set cover instance (U, S) as follows: the universe U is V, and the family of subsets is S = {S1, S2, ..., Sn} such that Sᵥ consists of the vertex v and all vertices adjacent to v in G. Now if D is a dominating set for G, then C = {Sv : v ∈ D} is a feasible solution of the set cover problem, with |C| = |D|. Conversely, if C = {Sv : v ∈ D} is a feasible solution of the set cover problem, then D is a dominating set for G, with |D| = |C|. Hence the size of a minimum dominating set for G equals the size of a minimum set cover for (U, S). Furthermore, there is a simple algorithm that maps a dominating set to a set cover of the same size and vice versa. In particular, an eﬃcient α-approximation algorithm for set covering provides an eﬃcient α-approximation algorithm for minimum dominating sets.

For example, given the graph G shown on the right, we construct a set cover instance with the universe U = {1, 2, ..., 6} and the subsets S1 = {1, 2, 5}, S2 = {1, 2, 3, 5}, S3 = {2, 3, 4, 6}, S4 = {3, 4}, S5 = {1, 2, 5, 6}, and S6 = {3, 5, 6}. In this example, D = {3, 5} is a dominating set for G – this corresponds to the set cover C = {S3, S5}. For example, the vertex 4 ∈ V is dominated by the vertex 3 ∈ D, and the element 4 ∈ U is contained in the set S3 ∈ C.

From set covering to dominating set. Let (S, U) be an instance of the set cover problem with the universe U and the family of subsets S = {Si : i ∈ I}; we assume that U and the index set I are disjoint. Construct a graph G = (V, E) 118 CHAPTER 31. DOMINATION PERFECT GRAPH 2 3 4

5 6

as follows: the set of vertices is V = I ∪ U, there is an edge {i, j} ∈ E between each pair i, j ∈ I, and there is also an edge {i, u} for each i ∈ I and u ∈ Si. That is, G is a split graph: I is a clique and U is an independent set. Now if C = {Si : i ∈ D} is a feasible solution of the set cover problem for some subset D ⊆ I, then D is a dominating set for G, with |D| = |C|: First, for each u ∈ U there is an i ∈ D such that u ∈ Si, and by construction, u and i are adjacent in G; hence u is dominated by i. Second, since D must be nonempty, each i ∈ I is adjacent to a vertex in D. Conversely, let D be a dominating set for G. Then it is possible to construct another dominating set X such that |X| ≤ |D| and X ⊆ I: simply replace each u ∈ D ∩ U by a neighbour i ∈ I of u. Then C = {Si : i ∈ X} is a feasible solution of the set cover problem, with |C| = |X| ≤ |D|.

The illustration on the right show the construction for U = {a, b, c, d, e}, I = {1, 2, 3, 4}, S1 = {a, b, c}, S2 = {a, b}, S3 = {b, c, d}, and S4 = {c, d, e}.

In this example, C = {S1, S4} is a set cover; this corresponds to the dominating set D = {1, 4}.

D = {a, 3, 4} is another dominating set for the graph G. Given D, we can construct a dominating set X = {1, 3, 4} which is not larger than D and which is a subset of I. The dominating set X corresponds to the set cover C = {S1, S3, S4}.

31.4.2 Special cases

If the graph has maximum degree Δ, then the greedy approximation algorithm finds an O(log Δ)-approximation of a minimum dominating set. Also, let d√be the cardinality of dominating set obtained using greedy approximation then following relation holds, d ≤ N+1 - 2M + 1 , where N is number of nodes and M is number of edges in given undirected graph.[2] For fixed Δ, this is qualifies Dominating Set for APX membership; in fact, it is APX-complete.[3] The problem admits a PTAS for special cases such as unit disk graphs and planar graphs (Crescenzi et al. 2000). A minimum dominating set can be found in linear time in series-parallel graphs (Takamizawa, Nishizeki & Saito 1982). 31.4. ALGORITHMS AND COMPUTATIONAL COMPLEXITY 119

23 4 1

a b c d e

31.4.3 Exact algorithms

A minimum dominating set of an n-vertex graph can be found in time O(2nn) by inspecting all vertex subsets. Fomin, Grandoni & Kratsch (2009) show how to ﬁnd a minimum dominating set in time O(1.5137n) and exponential space, and in time O(1.5264n) and polynomial space. A faster algorithm, using O(1.5048n) time was found by van Rooij, Nederlof & van Dijk (2009), who also show that the number of minimum dominating sets can be computed in this time. The number of minimal dominating sets is at most 1.7159n and all such sets can be listed in time O(1.7159n) (Fomin et al. 2008).

31.4.4 Parameterized complexity

Finding a dominating set of size k plays a central role in the theory of parameterized complexity. It is the most well- known problem complete for the class W[2] and used in many reductions to show intractability of other problems. In particular, the problem is not fixed-parameter tractable in the sense that no algorithm with running time f(k)nO(1) for any function f exists unless the W-hierarchy collapses to FPT=W[2]. On the other hand, if the input graph is planar, the problem remains NP-hard, but a fixed-parameter algorithm is known. In fact, the problem has a kernel of size linear in k (Alber, Fellows & Niedermeier 2004), and running times that are exponential in √k and cubic in n may be obtained by applying dynamic programming to a branch- decomposition of the kernel (Fomin & Thilikos 2006). More generally, the dominating set problem and many variants of the problem are fixed-parameter tractable when parameterized by both the size of the dominating set and the size of the smallest forbidden complete bipartite subgraph; that is, the problem is FPT on biclique-free graphs, a very general class of sparse graphs that includes the planar graphs (Telle & Villanger 2012). 120 CHAPTER 31. DOMINATION PERFECT GRAPH

31.5 Variants

Vizing’s conjecture relates the domination number of a cartesian product of graphs to the domination number of its factors. There has been much work on connected dominating sets. If S is a connected dominating set, one can form a spanning tree of G in which S forms the set of non-leaf vertices of the tree; conversely, if T is any spanning tree in a graph with more than two vertices, the non-leaf vertices of T form a connected dominating set. Therefore, finding minimum connected dominating sets is equivalent to finding spanning trees with the maximum possible number of leaves. A total dominating set is a set of vertices such that all vertices in the graph (including the vertices in the dominating set themselves) have a neighbor in the dominating set. Figure (c) above shows a dominating set that is a connected dominating set and a total dominating set; the examples in figures (a) and (b) are neither. A k-tuple dominating set is a set of vertices such that each vertex in the graph has at least k neighbors in the set. An (1+log n)-approximation of a minimum k-tuple dominating set can be found in polynomial time (Klasing & Laforest 2004). Similarly, a k-dominating set is a set of vertices such that each vertex not in the set has at least k neighbors in the set. While every graph admits a k-dominating set, only graphs with minimum degree k-1 admit a k-tuple dominating set. However, even if the graph admits k-tuple dominating set, a minimum k-tuple dominating set can be nearly k times larger than a minimum k-dominating set for the same graph (Förster 2013); An (1.7+log Δ)-approximation of a minimum k-dominating set can be found in polynomial time as well. A domatic partition is a partition of the vertices into disjoint dominating sets. The domatic number is the maximum size of a domatic partition. An Eternal dominating set is a dynamic version of domination in which a vertex v in dominating set D is chosen and replaced with a neighbor u (u is not in D) such that the modified D is also a dominating set and this process can be repeated over any infinite sequence of choices of vertices v.

31.6 Software for searching minimum dominating set

31.7 See also

• Set cover problem

• Bondage number

31.8 References

[1] Escoﬃer, Bruno; Paschos, Vangelis Th. (2006). “Completeness in approximation classes beyond APX”. Theoretical Computer Science 359 (1-3): 369–377. doi:10.1016/j.tcs.2006.05.023.

[2] Parekh, Abhay K. (1991). “Analysis of a greedy heuristic for ﬁnding small dominating sets in graphs”. Information Pro- cessing Letters 39 (5): 237–240. doi:10.1016/0020-0190(91)90021-9.

[3] Papadimitriou, Christos H.; Yannakakis, Mihailis (1991). “Optimization, Approximation, and Complexity Classes”. Jour- nal of Computer and Systems Sciences 43 (3): 425–440. doi:10.1016/0022-0000(91)90023-X.

• Alber, Jochen; Fellows, Michael R; Niedermeier, Rolf (2004), “Polynomial-time data reduction for dominating set”, Journal of the ACM 51 (3): 363–384, doi:10.1145/990308.990309.

• Allan, Robert B.; Laskar, Renu (1978), “On domination and independent domination numbers of a graph”, Discrete Mathematics 23 (2): 73–76, doi:10.1016/0012-365X(78)90105-X.

• Crescenzi, Pierluigi; Kann, Viggo; Halldórsson, Magnús; Karpinski, Marek; Woeginger, Gerhard (2000), “Minimum dominating set”, A Compendium of NP Optimization Problems.

• Faudree, Ralph; Flandrin, Evelyne; Ryjáček, Zdeněk (1997), “Claw-free graphs — A survey”, Discrete Math- ematics 164 (1–3): 87–147, doi:10.1016/S0012-365X(96)00045-3, MR 1432221. 31.8. REFERENCES 121

• Fomin, Fedor V.; Grandoni, Fabrizio; Kratsch, Dieter (2009), “A measure & conquer approach for the analysis of exact algorithms”, Journal of the ACM 56 (5): 25:1–32, doi:10.1145/1552285.1552286. • Fomin, Fedor V.; Grandoni, Fabrizio; Pyatkin, Artem; Stepanov, Alexey (2008), “Combinatorial bounds via measure and conquer: Bounding minimal dominating sets and applications”, ACM Transactions on Algorithms 5 (1): 9:1–17, doi:10.1145/1435375.1435384.

• Fomin, Fedor V.; Thilikos, Dimitrios M. (2006), “Dominating sets in planar graphs: branch-width and exponential speed-up”, SIAM Journal on Computing 36 (2): 281, doi:10.1137/S0097539702419649.

• Förster, Klaus-Tycho. (2013), “Approximating Fault-Tolerant Domination in General Graphs”, Proc. of the Tenth Workshop on Analytic Algorithmics and Combinatorics ANALCO, SIAM, pp. 25–32, doi:10.1137/1.9781611973037.4, ISBN 978-1-61197-254-2. • Garey, Michael R.; Johnson, David S. (1979), Computers and Intractability: A Guide to the Theory of NP- Completeness, W. H. Freeman, ISBN 0-7167-1045-5, p. 190, problem GT2.

• Grandoni, F. (2006), “A note on the complexity of minimum dominating set”, Journal of Discrete Algorithms 4 (2): 209–214, doi:10.1016/j.jda.2005.03.002.

• Guha, S.; Khuller, S. (1998), “Approximation algorithms for connected dominating sets”, Algorithmica 20 (4): 374–387, doi:10.1007/PL00009201.

• Haynes, Teresa W.; Hedetniemi, Stephen; Slater, Peter (1998a), Fundamentals of Domination in Graphs, Marcel Dekker, ISBN 0-8247-0033-3, OCLC 37903553.

• Haynes, Teresa W.; Hedetniemi, Stephen; Slater, Peter (1998b), Domination in Graphs: Advanced Topics, Marcel Dekker, ISBN 0-8247-0034-1, OCLC 38201061.

• Hedetniemi, S. T.; Laskar, R. C. (1990), “Bibliography on domination in graphs and some basic deﬁnitions of domination parameters”, Discrete Mathematics 86 (1–3): 257–277, doi:10.1016/0012-365X(90)90365-O.

• Klasing, Ralf; Laforest, Christian (2004), “Hardness results and approximation algorithms of k-tuple domination in graphs”, Information Processing Letters 89 (2): 75–83, doi:10.1016/j.ipl.2003.10.004.

• Kann, Viggo (1992), On the Approximability of NP-complete Optimization Problems (PDF). PhD thesis, De- partment of Numerical Analysis and Computing Science, Royal Institute of Technology, Stockholm.

• Raz, R.; Safra, S. (1997), “A sub-constant error-probability low-degree test, and sub-constant error-probability PCP characterization of NP”, Proc. 29th Annual ACM Symposium on Theory of Computing, ACM, pp. 475– 484, doi:10.1145/258533.258641, ISBN 0-89791-888-6. • Takamizawa, K.; Nishizeki, T.; Saito, N. (1982), “Linear-time computability of combinatorial problems on series-parallel graphs”, Journal of the ACM 29 (3): 623–641, doi:10.1145/322326.322328. • Telle, Jan Arne; Villanger, Yngve (2012), “FPT algorithms for domination in biclique-free graphs”, in Ep- stein, Leah; Ferragina, Paolo, Algorithms – ESA 2012: 20th Annual European Symposium, Ljubljana, Slove- nia, September 10–12, 2012, Proceedings, Lecture Notes in Computer Science 7501, Springer, pp. 802–812, doi:10.1007/978-3-642-33090-2_69.

• van Rooij, J. M. M.; Nederlof, J.; van Dijk, T. C. (2009), “Inclusion/Exclusion Meets Measure and Conquer: Exact Algorithms for Counting Dominating Sets”, Proc. 17th Annual European Symposium on Algorithms, ESA 2009, Lecture Notes in Computer Science 5757, Springer, pp. 554–565, doi:10.1007/978-3-642-04128-0_50, ISBN 978-3-642-04127-3. Chapter 32

Dually chordal graph

In the mathematical area of graph theory, an undirected graph G is dually chordal if the hypergraph of its maximal cliques is a hypertree. The name comes from the fact that a graph is chordal if and only if the hypergraph of its maximal cliques is the dual of a hypertree. Originally, these graphs were defined by maximum neighborhood orderings and have a variety of different characterizations - see e.g. [1] Unlike for chordal graphs, the property of being dually chordal is not hereditary, i.e., induced subgraphs of a dually chordal graph are not necessarily dually chordal (hereditarily dually chordal graphs are exactly the strongly chordal graphs), and a dually chordal graph is in general not a perfect graph. Dually chordal graphs appeared first under the name HT-graphs in.[2]

32.1 Characterizations

Dually chordal graphs are the clique graphs of chordal graphs,[3] i.e., the intersection graphs of maximal cliques of chordal graphs. The following properties are equivalent: [4] (1) G has a maximum neighborhood ordering. (2) There is a spanning tree T of G such that any maximal clique of G induces a subtree in T. (3) There is a spanning tree T of G such that any maximal clique of G induces a subtree in T. (4) The closed neighborhood hypergraph N(G) of G is a hypertree. (5) The maximal clique hypergraph of G is a hypertree. (6) G is the 2-section graph of a hypertree. Condition (4) also implies that a graph is dually chordal if and only if its square is chordal and its closed neighborhood hypergraph has the Helly property. In [5] dually chordal graphs are characterized in terms of separator properties. In [6] it was shown that dually chordal graphs are precisely the intersection graphs of maximal hypercubes of graphs of acyclic cubical complexes. The structure and algorithmic use of doubly chordal graphs is given by Moscarini in [7] - these are graphs which are chordal and dually chordal.

32.2 Recognition

Dually chordal graphs can be recognized in linear time, and a maximum neighborhood ordering of a dually chordal graph can be found in linear time. [8]

32.3 Complexity of problems

While some basic problems such as maximum independent set, maximum clique, coloring and clique cover remain NP-complete for dually chordal graphs, some variants of the minimum dominating set problem and Steiner tree are efficiently solvable on dually chordal graphs (but Independent Domination remains NP-complete). [9] See [10] for the use of dually chordal graph properties for tree spanners, and see [11] for a linear time algorithm of efficient domination and efficient edge domination on dually chordal graphs.

122 32.4. NOTES 123

32.4 Notes

[1] Brandstädt et al. (1993); Brandstädt, Chepoi & Dragan (1994); Brandstädt et al. (1994); Brandstädt et al. (1998); Brandstädt, Le & Spinrad (1999)

[2] Dragan (1989); Dragan, Prisacaru & Chepoi (1992); Dragan (1993)

[3] Gutierrez & Oubina (1996);Szwarcﬁter & Bornstein (1996)

[4] Brandstädt et al. (1993);Brandstädt, Chepoi & Dragan (1998);Brandstädt et al. (1998); Dragan, Prisacaru & Chepoi (1992); Dragan (1993)

[5] de Caria & Gutierrez (2012)

[6] Brešar (2003)

[7] Moscarini (1993)

[8] Brandstädt, Chepoi & Dragan (1998);Dragan (1989)

[9] Brandstädt, Chepoi & Dragan (1998); Dragan, Prisacaru & Chepoi (1992); Dragan (1993)

[10] Brandstädt, Chepoi & Dragan (1999)

[11] Brandstädt, Leitert & Rautenbach (2012)

32.5 References

• Brandstädt, Andreas; Chepoi, Victor; Dragan, Feodor (1998), “The algorithmic use of hypertree structure and maximum neighborhood orderings”, Discrete Applied Mathematics 82: 43–77. • Brandstädt, Andreas; Chepoi, Victor; Dragan, Feodor (1999), “Distance approximating trees for chordal and dually chordal graphs”, Journal of Algorithms 30: 166–184. • Brandstädt, Andreas; Dragan, Feodor; Chepoi, Victor; Voloshin, Vitaly (1993), “Dually Chordal Graphs”, Lecture Notes in Computer Science 790: 237–251. • Brandstädt, Andreas; Dragan, Feodor; Chepoi, Victor; Voloshin, Vitaly (1998), “Dually Chordal Graphs”, SIAM Journal on Discrete Mathematics 11: 437–455. • Brandstädt, Andreas; Le, Van Bang; Spinrad, Jeremy (1999), Graph Classes: A Survey, SIAM Monographs on Discrete Mathematics and Applications, ISBN 0-89871-432-X. • Brandstädt, Andreas; Leitert, Arne; Rautenbach, Dieter (2012), “Eﬃcient Dominating and Edge Dominating Sets for Graphs and Hypergraphs”, Lecture Notes in Computer Science 7676: 267–277. • Brešar, Boštjan (2003), “Intersection graphs of maximal hypercubes”, European Journal of Combinatorics 24: 195–209. • de Caria, Pablo; Gutierrez, Marisa (2012), “On Minimal Vertex Separators of Dually Chordal Graphs: Prop- erties and Characterizations”, Discrete Applied Mathematics 160: 2627–2635. • Dragan, Feodor (1989), Centers of Graphs and the Helly Property (in Russian), Ph.D. thesis, Moldova State University, Moldova. • Dragan, Feodor; Prisacaru, Chiril; Chepoi, Victor (1992), “Location problems in graphs and the Helly property (in Russian)", Discrete Math. (Moscow) 4: 67–73. • Dragan, Feodor (1993), “HT-graphs: centers, connected r-domination and Steiner trees”, Comput. Sci. J. of Moldova (Kishinev) 1: 64–83. • Gutierrez, Marisa; Oubina, L. (1996), “Metric Characterizations of proper Interval Graphs and Tree-Clique Graphs”, Journal of Graph Theory 21: 199–205. • McKee, Terry A.; McMorris, FR. (1999), Topics in Intersection Graph Theory, SIAM Monographs on Discrete Mathematics and Applications. 124 CHAPTER 32. DUALLY CHORDAL GRAPH

• Moscarini, Marina (1993), “Doubly Chordal Graphs, Steiner trees and connected domination”, Networks 23: 59–69. • Szwarcﬁter, Jayme L.; Bornstein, Claudson F. (1994), “Clique Graphs of Chordal and Path Graphs”, SIAM Journal on Discrete Mathematics 7: 331–336. Chapter 33

Edge-transitive graph

This article is about graph theory. For edge transitivity in geometry, see Edge-transitive.

In the mathematical ﬁeld of graph theory, an edge-transitive graph is a graph G such that, given any two edges e1 [1] and e2 of G, there is an automorphism of G that maps e1 to e2. In other words, a graph is edge-transitive if its automorphism group acts transitively upon its edges.

33.1 Examples and properties

Edge-transitive graphs include any complete bipartite graph Km,n , and any symmetric graph, such as the vertices and edges of the cube.[1] Symmetric graphs are also vertex-transitive (if they are connected), but in general edge- transitive graphs need not be vertex-transitive. The Gray graph is an example of a graph which is edge-transitive but not vertex-transitive. All such graphs are bipartite,[1] and hence can be colored with only two colors. An edge-transitive graph that is also regular, but not vertex-transitive, is called semi-symmetric. The Gray graph again provides an example. Every edge-transitive graph that is not vertex-transitive must be bipartite and either semi-symmetric or biregular.[2]

33.2 See also

• Edge-transitive (in geometry)

33.3 References

[1] Biggs, Norman (1993). Algebraic Graph Theory (2nd ed.). Cambridge: Cambridge University Press. p. 118. ISBN 0-521-45897-8.

[2] Lauri, Josef; Scapellato, Raﬀaele (2003), Topics in Graph Automorphisms and Reconstruction, London Mathematical So- ciety Student Texts, Cambridge University Press, pp. 20–21, ISBN 9780521529037.

33.4 External links

• Weisstein, Eric W., “Edge-transitive graph”, MathWorld.

125 126 CHAPTER 33. EDGE-TRANSITIVE GRAPH

The Gray graph is edge-transitive and regular, but not vertex-transitive. Chapter 34

Even-hole-free graph

In the mathematical area of graph theory, a graph is even-hole-free if it contains no induced cycle with an even number of vertices. Addario-Berry et al. (2008) demonstrated that every even-hole-free graph contains a bisimplicial vertex, which settled a conjecture by Reed.

34.1 Recognition

Conforti et al. (2002) gave the ﬁrst polynomial time recognition algorithm for even-hole-free graphs, which runs in O(n40) time.[1] da Silva & Vušković (2008) later improved this to O(n19) . The best currently known algorithm is given by Chang & Lu (2012) and Chang & Lu (2015) which runs in O(n11) time. While even-hole-free graphs can be recognized in polynomial time, it is NP-complete to determine whether a graph contains an even hole that includes a speciﬁc vertex.[2]

34.2 Notes

[1] Conforti et al. (2002) present their algorithm and assert that it runs in polynomial time without giving an explicit analysis. Chudnovsky, Kawarabayashi & Seymour (2004) estimate that it runs in “time about O(n40) .”

[2] Bienstock (1991)

34.3 References

• Addario-Berry, Louigi; Chudnovsky, Maria; Havet, Frédéric; Reed, Bruce; Seymour, Paul (2008), “Bisim- plicial vertices in even-hole-free graphs”, Journal of Combinatorial Theory, Series B 98 (6): 1119–1164, doi:10.1016/j.jctb.2007.12.006 • Bienstock, Dan (1991), “On the complexity of testing for odd holes and induced odd paths”, Discrete Mathe- matics 90 (1): 85–92, doi:10.1016/0012-365X(91)90098-M • Chudnovsky, Maria; Kawarabayashi, Ken-ichi; Seymour, Paul (2004), “Detecting even holes”, Journal of Graph Theory 48 (2): 85–111, doi:10.1002/jgt.20040 • Conforti, Michele; Cornuéjols, Gérard; Kapoor, Ajai; Vušković, Kristina (2002), “Even-hole-free graphs part I: Decomposition theorem”, Journal of Graph Theory 39 (1): 6–49, doi:10.1002/jgt.10006 • Conforti, Michele; Cornuéjols, Gérard; Kapoor, Ajai; Vušković, Kristina (2002), “Even-hole-free graphs part II: Recognition algorithm”, Journal of Graph Theory 40 (4): 238–266, doi:10.1002/jgt.10045 • da Silva, Murilo V.G.; Vušković, Kristina (2008), Decomposition of even-hole-free graphs with star cutsets and 2-joins

127 128 CHAPTER 34. EVEN-HOLE-FREE GRAPH

• Chang, Hsien-Chih; Lu, Hsueh-I (2012), “A Faster Algorithm to Recognize Even-Hole-Free Graphs”, SODA

• Chang, Hsien-Chih; Lu, Hsueh-I (2015), “A Faster Algorithm to Recognize Even-Hole-Free Graphs”, Journal of Combinatorial Theory, Series B 113: 141–161, doi:10.1016/j.jctb.2015.02.001

34.4 External links

• “Graphclass: even-cycle-free”. Information System on Graph Classes and their Inclusions. External link in |work= (help) Chapter 35

Expander graph

In combinatorics, an expander graph is a sparse graph that has strong connectivity properties, quantiﬁed using vertex, edge or spectral expansion as described below. Expander constructions have spawned research in pure and applied mathematics, with several applications to complexity theory, design of robust computer networks, and the theory of error-correcting codes.[1]

35.1 Deﬁnitions

Intuitively, an expander is a finite, undirected multigraph in which every subset of the vertices that is not “too large” has a “large” boundary. Different formalisations of these notions give rise to different notions of expanders: edge expanders, vertex expanders, and spectral expanders, as defined below. A disconnected graph is not an expander, since the boundary of a connected component is empty. Every connected graph is an expander; however, different connected graphs have different expansion parameters. The complete graph has the best expansion property, but it has largest possible degree. Informally, a graph is a good expander if it has low degree and high expansion parameters.

35.1.1 Edge expansion

The edge expansion (also isoperimetric number or Cheeger constant) h(G) of a graph G on n vertices is deﬁned as

|∂S| h(G) = min , | |≤ n | | 0< S 2 S

where ∂S := {(u, v) ∈ E(G): u ∈ S, v ∈ V (G) \ S}. In the equation, the minimum is over all nonempty sets S of at most n/2 vertices and ∂S is the edge boundary of S, i.e., the set of edges with exactly one endpoint in S.[2]

35.1.2 Vertex expansion

The vertex isoperimetric number hout(G) (also vertex expansion or magniﬁcation) of a graph G is deﬁned as

|∂out(S)| hout(G) = min , | |≤ n | | 0< S 2 S

[3] where ∂out(S) is the outer boundary of S, i.e., the set of vertices in V (G) \ S with at least one neighbor in S. In a variant of this deﬁnition (called unique neighbor expansion) ∂out(S) is replaced by the set of vertices in V with exactly one neighbor in S.[4]

129 130 CHAPTER 35. EXPANDER GRAPH

The vertex isoperimetric number hin(G) of a graph G is deﬁned as

|∂in(S)| hin(G) = min , | |≤ n | | 0< S 2 S

[3] where ∂in(S) is the inner boundary of S, i.e., the set of vertices in S with at least one neighbor in V (G) \ S .

35.1.3 Spectral expansion

When G is d-regular, a linear algebraic deﬁnition of expansion is possible based on the eigenvalues of the adjacency [5] matrix A = A(G) of G, where Aij is the number of edges between vertices i and j. Because A is symmetric, the spectral theorem implies that A has n real-valued eigenvalues λ1 ≥ λ2 ≥ · · · ≥ λn . It is known that all these eigenvalues are in [−d, d]. n Because G is regular, the uniform distribution u ∈ R with ui = 1/n for all i = 1, ..., n is the stationary distribution of G. That is, we have Au = du, and u is an eigenvector of A with eigenvalue λ1 = d, where d is the degree of the [6] vertices of G. The spectral gap of G is deﬁned to be d−λ2, and it measures the spectral expansion of the graph G. It is known that λn = −d if and only if G is bipartite. In many contexts, for example in the expander mixing lemma, a bound on λ2 is not enough, but indeed it is necessary to bound the absolute value of all the eigenvalues away from d:

λ = max{|λ2|, |λn|}

Since this is the largest eigenvalue corresponding to an eigenvector orthogonal to u, it can be equivalently deﬁned using the Rayleigh quotient:

∥Av∥ λ = max 2 , v⊥u,v=0̸ ∥v∥2

where

( ) ∑n 1/2 ∥ ∥ 2 v 2 = vi i=1

is the 2-norm of the vector v ∈ Rn . The normalized versions of these deﬁnitions are also widely used and more convenient in stating some results. Here 1 one considers the matrix d A , which is the Markov transition matrix of the graph G. Its eigenvalues are between −1 and 1. For not necessarily regular graphs, the spectrum of a graph can be deﬁned similarly using the eigenvalues of the Laplacian matrix. For directed graphs, one considers the singular values of the adjacency matrix A, which are equal to the roots of the eigenvalues of the symmetric matrix ATA.

35.2 Relationships between diﬀerent expansion properties

The expansion parameters deﬁned above are related to each other. In particular, for any d-regular graph G,

hout(G) ≤ h(G) ≤ d · hout(G).

Consequently, for constant degree graphs, vertex and edge expansion are qualitatively the same. 35.3. CONSTRUCTIONS 131

35.2.1 Cheeger inequalities

When G is d-regular, there is a relationship between h(G) and the spectral gap d − λ2 of G. An inequality due to Tanner and independently Alon and Milman[7] states that[8]

√ 1 − ≤ ≤ − 2 (d λ2) h(G) 2d(d λ2). This inequality is closely related to the Cheeger bound for Markov chains and can be seen as a discrete version of Cheeger’s inequality in Riemannian geometry. Similar connections between vertex isoperimetric numbers and the spectral gap have also been studied:[9]

(√ )2 hout(G) ≤ 4(d − λ2) + 1 − 1 √ hin(G) ≤ 8(d − λ2). h2 2 − Asymptotically speaking, the quantities d , hout , and hin are all bounded above by the spectral gap O(d λ2) .

35.3 Constructions

There are three general strategies for constructing families of expander graphs.[10] The ﬁrst strategy is algebraic and group-theoretic, the second strategy is analytic and uses additive combinatorics, and the third strategy is combinatorial and uses the zig-zag and related graphs products. Noga Alon showed that certain graphs constructed from ﬁnite geometries are the sparsest examples of highly expanding graphs.[11]

35.3.1 Margulis-Gabber-Galil

Algebraic constructions based on Cayley graphs are known for various variants of expander graphs. The following construction is due to Margulis and has been analysed by Gabber and Galil.[12] For every natural number n, one considers the graph Gn with the vertex set Zn × Zn , where Zn = Z/nZ : For every vertex (x, y) ∈ Zn × Zn , its eight adjacent vertices are

(x 2y, y), (x (2y + 1), y), (x, y 2x), (x, y (2x + 1)). Then the following holds: √ Theorem. For all n, the graph Gn has second-largest eigenvalue λ(G) ≤ 5 2 .

35.3.2 Ramanujan graphs

Main article: Ramanujan graph √ By a theorem of Alon and Boppana, all large enough d-regular graphs satisfy λ ≥ 2 d − 1 − o(1) , where λ is the second largest eigenvalue in√ absolute value.[13] Ramanujan graphs are d-regular graphs for which this bound is tight. That is, they satisfy λ ≤ 2 d − 1 .[14] Hence Ramanujan graphs have an asymptotically smallest possible λ. They are also excellent spectral expanders. Lubotzky, Phillips, and Sarnak (1988), Margulis (1988), and Morgenstern (1994) show how Ramanujan graphs can be constructed explicitly.[15] By a theorem of Friedman (2003), random d-regular graphs on n vertices are almost Ramanujan, that is, they satisfy

√ λ ≤ 2 d − 1 + ϵ with probability 1 − o(1) as n → ∞ tends to inﬁnity.[16] 132 CHAPTER 35. EXPANDER GRAPH

35.4 Applications and useful properties

The original motivation for expanders is to build economical robust networks (phone or computer): an expander with bounded valence is precisely an asymptotic robust graph with number of edges growing linearly with size (number of vertices), for all subsets. Expander graphs have found extensive applications in computer science, in designing algorithms, error correcting codes, extractors, pseudorandom generators, sorting networks (Ajtai, Komlós & Szemerédi (1983)) and robust computer networks. They have also been used in proofs of many important results in computational complexity theory, such as SL=L (Reingold (2008)) and the PCP theorem (Dinur (2007)). In cryptography, expander graphs are used to construct hash functions. The following are some properties of expander graphs that have proven useful in many areas.

35.4.1 Expander mixing lemma

Main article: Expander mixing lemma

The expander mixing lemma states that, for any two subsets of the vertices S, T ⊆ V, the number of edges between S and T is approximately what you would expect in a random d-regular graph. The approximation is better the smaller λ = max{|λ2|, |λn|} is. In a random d-regular graph, as well as in an Erdős–Rényi random graph with edge d · | | · | | probability d/n, we expect n S T edges between S and T. More formally, let E(S, T) denote the number of edges between S and T. If the two sets are not disjoint, edges in their intersection are counted twice, that is,

E(S, T ) = 2|E(G[S ∩ T ])| + E(S \ T,T ) + E(S, T \ S)

Then the expander mixing lemma says that the following inequality holds:

√ d · |S| · |T | E(S, T ) − ≤ dλ |S| · |T |, n where λ is the absolute value of the normalized second largest eigenvalue.

35.4.2 Expander walk sampling

Main article: Expander walk sampling

The Chernoﬀ bound states that, when sampling many independent samples from a random variables in the range [−1, 1], with high probability the average of our samples is close to the expectation of the random variable. The expander walk sampling lemma, due to Ajtai, Komlós & Szemerédi (1987) and Gillman (1998), states that this also holds true when sampling from a walk on an expander graph. This is particularly useful in the theory of derandomization, since sampling according to an expander walk uses many fewer random bits than sampling independently.

35.5 See also

• Algebraic connectivity

• Zig-zag product

• Superstrong approximation 35.6. NOTES 133

35.6 Notes

[1] Hoory, Linial & Widgerson (2006)

[2] Deﬁnition 2.1 in Hoory, Linial & Widgerson (2006)

[3] Bobkov, Houdré & Tetali (2000)

[4] Alon & Capalbo (2002)

[5] cf. Section 2.3 in Hoory, Linial & Widgerson (2006)

[6] This deﬁnition of the spectral gap is from Section 2.3 in Hoory, Linial & Widgerson (2006)

[7] Alon & Spencer 2011.

[8] Theorem 2.4 in Hoory, Linial & Widgerson (2006)

[9] See Theorem 1 and p.156, l.1 in Bobkov, Houdré & Tetali (2000). Note that λ2 there corresponds to 2(d − λ2) of the current article (see p.153, l.5)

[10] see, e.g., Yehudayoﬀ (2012)

[11] http://link.springer.com/article/10.1007%2FBF02579382

[12] see, e.g., p.9 of Goldreich (2011)

[13] Theorem 2.7 of Hoory, Linial & Widgerson (2006)

[14] Deﬁnition 5.11 of Hoory, Linial & Widgerson (2006)

[15] Theorem 5.12 of Hoory, Linial & Widgerson (2006)

[16] Theorem 7.10 of Hoory, Linial & Widgerson (2006)

35.7 References

Textbooks and surveys

• Alon, N.; Spencer, Joel H. (2011). “9.2. Eigenvalues and Expanders”. The Probabilistic Method (3rd ed.). John Wiley & Sons.

• Chung, Fan R. K. (1997), Spectral Graph Theory, CBMS Regional Conference Series in Mathematics 92, American Mathematical Society, ISBN 0-8218-0315-8

• Davidoﬀ, Guiliana; Sarnak, Peter; Valette, Alain (2003), Elementary number theory, group theory and Ra- manujan graphs, LMS student texts 55, Cambridge University Press, ISBN 0-521-53143-8

• Hoory, Shlomo; Linial, Nathan; Widgerson, Avi (2006), “Expander graphs and their applications” (PDF), Bulletin (New series) of the American Mathematical Society 43 (4): 439–561, doi:10.1090/S0273-0979-06- 01126-8

• Krebs, Mike; Shaheen, Anthony (2011), Expander families and Cayley graphs: A beginner’s guide, Oxford University Press, ISBN 0-19-976711-4

Research articles

• Ajtai, M.; Komlós, J.; Szemerédi, E. (1983), “An O(n log n) sorting network”, Proceedings of the 15th Annual ACM Symposium on Theory of Computing, pp. 1–9, doi:10.1145/800061.808726, ISBN 0-89791-099-0

• Ajtai, M.; Komlós, J.; Szemerédi, E. (1987), “Proceedings of the 19th Annual ACM Symposium on Theory of Computing”, ACM, pp. 132–140, doi:10.1145/28395.28410, ISBN 0-89791-221-7 |chapter= ignored (help) 134 CHAPTER 35. EXPANDER GRAPH

• Alon, N.; Capalbo, M. (2002). “Explicit unique-neighbor expanders”. The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings. p. 73. doi:10.1109/SFCS.2002.1181884. ISBN 0-7695-1822-2.

• Bobkov, S.; Houdré, C.; Tetali, P. (2000), "λ∞, vertex isoperimetry and concentration”, Combinatorica 20 (2): 153–172, doi:10.1007/s004930070018.

• Dinur, Irit (2007), “The PCP theorem by gap ampliﬁcation”, Journal of the ACM 54 (3): 12, doi:10.1145/1236457.1236459. • Gillman, D. (1998), “A Chernoﬀ Bound for Random Walks on Expander Graphs”, SIAM Journal on Computing (Society for Industrial and Applied Mathematics) 27 (4,): 1203–1220, doi:10.1137/S0097539794268765 • Goldreich, Oded (2011), “Basic Facts about Expander Graphs”, Studies in Complexity and Cryptography: 451– 464, doi:10.1007/978-3-642-22670-0_30 • Reingold, Omer (2008), “Undirected connectivity in log-space”, Journal of the ACM 55 (4): Article 17, 24 pages, doi:10.1145/1391289.1391291

• Yehudayoﬀ, Amir (2012), “Proving expansion in three steps”, ACM SIGACT News 43 (3): 67–84, doi:10.1145/2421096.2421115

35.8 External links

• Brief introduction in Notices of the American Mathematical Society

• Introductory paper by Michael Nielsen • Lecture notes from a course on expanders (by Nati Linial and Avi Wigderson)

• Lecture notes from a course on expanders (by Prahladh Harsha) • Deﬁnition and application of spectral gap Chapter 36

Extractor (mathematics)

An (N, M, D, K, ϵ) -extractor is a bipartite graph with N nodes on the left and M nodes on the right such that each node on the left has D neighbors (on the right), which has the added property that for any subset A of the left vertices of size at least K , the distribution on right vertices obtained by choosing a random node in A and then following a random edge to get a node x on the right side is ϵ -close to the uniform distribution in terms of total variation distance. A disperser is a related graph. An equivalent way to view an extractor is as a bivariate function

E :[N] × [D] → [M]

in the natural way. With this view it turns out that the extractor property is equivalent to: for any source of randomness X that gives n bits with min-entropy log K , the distribution E(X,UD) is ϵ -close to UM , where UT denotes the uniform distribution on [T ] . Extractors are interesting when they can be constructed with small K, D, ϵ relative to N and M is as close to KD (the total randomness in the input sources) as possible. Extractor functions were originally researched as a way to extract randomness from weakly random sources. See randomness extractor. Using the probabilistic method it is easy to show that extractor graphs with really good parameters exist. The challenge is to ﬁnd explicit or polynomial time computable examples of such graphs with good parameters. Algorithms that compute extractor (and disperser) graphs have found many applications in computer science.

36.1 References

• Ronen Shaltiel, Recent developments in extractors - a survey

135 Chapter 37

Factor-critical graph

A factor-critical graph, together with perfect matchings of the subgraphs formed by removing one of its vertices.

In graph theory, a mathematical discipline, a factor-critical graph (or hypomatchable graph[1][2]) is a graph with n vertices in which every subgraph of n − 1 vertices has a perfect matching. (A perfect matching in a graph is a subset of its edges with the property that each of its vertices is the endpoint of exactly one of the edges in the subset.) A matching that covers all but one vertex of a graph is called a near-perfect matching. So equivalently, a factor- critical graph is a graph in which there are near-perfect matchings that avoid every possible vertex.

136 37.1. EXAMPLES 137

37.1 Examples

Three friendship graphs, examples of non-Hamiltonian factor-critical graphs

The gyroelongated pentagonal pyramid, a claw-free factor-critical graph

Any odd-length cycle graph is factor-critical,[1] as is any complete graph with an odd number of vertices.[3] More generally, every Hamiltonian graph with an odd number of vertices is factor-critical. The friendship graphs (graphs formed by connecting a collection of triangles at a single common vertex) provide examples of graphs that are factor- critical but not Hamiltonian. If a graph G is factor-critical, then so is the Mycielskian of G. For instance, the Grötzsch graph, the Mycielskian of a ﬁve-vertex cycle-graph, is factor-critical.[4] Every 2-vertex-connected claw-free graph with an odd number of vertices is factor-critical. For instance, the 11- vertex graph formed by removing a vertex from the regular icosahedron (the graph of the gyroelongated pentagonal 138 CHAPTER 37. FACTOR-CRITICAL GRAPH

pyramid) is both 2-connected and claw-free, so it is factor-critical. This result follows directly from the more fundamental theorem that every connected claw-free graph with an even number of vertices has a perfect matching.[5]

37.2 Characterizations

Factor-critical graphs may be characterized in several diﬀerent ways, other than their deﬁnition as graphs in which each vertex deletion allows for a perfect matching:

• Tibor Gallai proved that a graph is factor-critical if and only if it is connected and, for each node v of the graph, there exists a maximum matching that does not include v. It follows from these properties that the graph must have an odd number of vertices and that every maximum matching must match all but one vertex.[6] • László Lovász proved that a graph is factor-critical if and only if it has an odd ear decomposition, a partition of its edges into a sequence of subgraphs, each of which is an odd-length path or cycle, with the first in the sequence being a cycle, each path in the sequence having both endpoints but no interior points on vertices in previous subgraphs, and each cycle other than the first in the sequence having exactly one vertex in previous subgraphs. For instance, the graph in the illustration may be partitioned in this way into a cycle of five edges and a path of three edges. In the case that a near-perfect matching of the factor-critical graph is also given, the ear decomposition can be chosen in such a way that each ear alternates between matched and unmatched edges.[7][8] • A graph is also factor-critical if and only if it can be reduced to a single graph by a sequence of contractions of odd-length cycles. Moreover, in this characterization, it is possible to choose each cycle in the sequence so that it contains the vertex formed by the contraction of the previous cycle.[1] For instance, if one contracts the ears of an ear decomposition, in the order given by the decomposition, then at the time each ear is contracted it forms an odd cycle, so the ear decomposition characterization may be used to find a sequence of odd cycles to contract. Conversely from a sequence of odd cycle contractions, each containing the vertex formed from the previous contraction, one may form an ear decomposition in which the ears are the sets of edges contracted in each step. • Suppose that a graph G is given together with a choice of a vertex v and a matching M that covers all vertices other than v. Then G is factor-critical if and only if there is a set of paths in G, alternating between matched and unmatched edges, that connect v to each of the other vertices in G. Based on this property, it is possible to determine in linear time whether a graph G with a given near-perfect matching is factor-critical.[9]

37.3 Properties

Factor-critical graphs must always have an odd number of vertices, and must be 2-edge-connected (that is, they cannot have any bridges).[10] However, they are not necessarily 2-vertex-connected; the friendship graphs provide a counterexample. It is not possible for a factor-critical graph to be bipartite, because in a bipartite graph with a near-perfect matching, the only vertices that can be deleted to produce a perfectly matchable graph are the ones on the larger side of the bipartition. Every 2-vertex-connected factor-critical graph with m edges has at least m different near-perfect matchings, and more generally every factor-critical graph with m edges and c blocks (2-vertex-connected components) has at least m − c + 1 different near-perfect matchings. The graphs for which these bounds are tight may be characterized by having odd ear decompositions of a specific form.[3] Any connected graph may be transformed into a factor-critical graph by contracting sufficiently many of its edges. The minimal sets of edges that need to be contracted to make a given graph G factor-critical form the bases of a matroid, a fact that implies that a greedy algorithm may be used to find the minimum weight set of edges to contract to make a graph factor-critical, in polynomial time.[11]

37.4 Applications

A blossom is a factor-critical subgraph of a larger graph. Blossoms play a key role in Jack Edmonds' algorithms for maximum matching and minimum weight perfect matching in non-bipartite graphs.[12] 37.5. GENERALIZATIONS AND RELATED CONCEPTS 139

In polyhedral combinatorics, factor-critical graphs play an important role in describing facets of the matching polytope of a given graph.[1][2]

37.5 Generalizations and related concepts

A graph is said to be k-factor-critical if every subset of n − k vertices has a perfect matching. Under this definition, a hypomatchable graph is 1-factor-critical.[13] Even more generally, a graph is (a,b)-factor-critical if every subset of n − k vertices has an r-factor, that is, it is the vertex set of an r-regular subgraph of the given graph. A critical graph (without qualification) is usually assumed to mean a graph for which removing each of its vertices reduces the number of colors it needs in a graph coloring. The concept of criticality has been used much more generally in graph theory to refer to graphs for which removing each possible vertex changes or does not change some relevant property of the graph. A matching-critical graph is a graph for which the removal of any vertex does not change the size of a maximum matching; by Gallai’s characterization, the matching-critical graphs are exactly the graphs in which every connected component is factor-critical.[14] The complement graph of a critical graph is necessarily matching-critical, a fact that was used by Gallai to prove lower bounds on the number of vertices in a critical graph.[15] Beyond graph theory, the concept of factor-criticality has been extended to matroids by defining a type of ear decomposition on matroids and defining a matroid to be factor-critical if it has an ear decomposition in which all ears are odd.[16]

37.6 References

[1] Pulleyblank, W. R.; Edmonds, J. (1974), “Facets of 1-matching polyhedra”, in Berge, C.; Ray-Chaudhuri, D. K., Hy- pergraph Seminar, Lecture Notes in Mathematics 411, Springer-Verlag, pp. 214–242, doi:10.1007/BFb0066196, ISBN 978-3-540-06846-4.

[2] Cornuéjols, G.; Pulleyblank, W. R. (1983), “Critical graphs, matchings and tours or a hierarchy of relaxations for the travelling salesman problem”, Combinatorica 3 (1): 35–52, doi:10.1007/BF02579340, MR 716420.

[3] Liu, Yan; Hao, Jianxiu (2002), “The enumeration of near-perfect matchings of factor-critical graphs”, Discrete Mathematics 243 (1–3): 259–266, doi:10.1016/S0012-365X(01)00204-7, MR 1874747.

[4] Došlić, Tomislav (2005), “Mycielskians and matchings”, Discussiones Mathematicae Graph Theory 25 (3): 261–266, MR 2232992.

[5] Favaron, Odile; Flandrin, Evelyne; Ryjáček, Zdeněk (1997), “Factor-criticality and matching extension in DCT-graphs”, Discussiones Mathematicae Graph Theory 17 (2): 271–278, MR 1627955.

[6] Gallai, T. (1963), “Neuer Beweis eines Tutte’schen Satzes”, Magyar Tud. Akad. Mat. Kutató Int. Közl. 8: 135–139, MR 0166777. As cited by Frank, András; Szegő, László (2002), “Note on the path-matching formula” (PDF), Journal of Graph Theory 41 (2): 110–119, doi:10.1002/jgt.10055, MR 1926313.

[7] Lovász, L. (1972), “A note on factor-critical graphs”, Studia Sci. Math. Hungar. 7: 279–280, MR 0335371.

[8] Korte, Bernhard H.; Vygen, Jens (2008), “10.4 Ear-Decompositions of Factor-Critical Graphs”, Combinatorial Optimiza- tion: Theory and Algorithms, Algorithms and Combinatorics 21 (4th ed.), Springer-Verlag, pp. 235–241, ISBN 978-3- 540-71843-7.

[9] Lou, Dingjun; Rao, Dongning (2004), “Characterizing factor critical graphs and an algorithm” (PDF), The Australasian Journal of Combinatorics 30: 51–56, MR 2080453.

[10] Seyﬀarth, Karen (1993), “Packings and perfect path double covers of maximal planar graphs”, Discrete Mathematics 117 (1–3): 183–195, doi:10.1016/0012-365X(93)90334-P, MR 1226141.

[11] Szigeti, Zoltán (1996), “On a matroid deﬁned by ear-decompositions of graphs”, Combinatorica 16 (2): 233–241, doi:10.1007/BF01844849, MR 1401896. For an earlier algorithm for contracting the minimum number of (unweighted) edges to reach a factor-critical graph, see Frank, András (1993), “Conservative weightings and ear-decompositions of graphs”, Combinatorica 13 (1): 65– 81, doi:10.1007/BF01202790, MR 1221177.

[12] Edmonds, Jack (1965), “Paths, Trees and Flowers”, Canadian Journal of Mathematics 17: 449–467, doi:10.4153/CJM- 1965-045-4, MR 0177907. 140 CHAPTER 37. FACTOR-CRITICAL GRAPH

[13] Favaron, Odile (1996), “On k-factor-critical graphs”, Discussiones Mathematicae Graph Theory 16 (1): 41–51, MR 1429805.

[14] Erdős, P.; Füredi, Z.; Gould, R. J.; Gunderson, D. S. (1995), “Extremal graphs for intersecting triangles”, Journal of Combinatorial Theory, Series B 64 (1): 89–100, doi:10.1006/jctb.1995.1026, MR 1328293.

[15] Gallai, T. (1963b), “Kritische Graphen II”, Publ. Math. Inst. Hungar. Acad. Sci. 8: 373–395. As cited by Stehlík, Matěj (2003), “Critical graphs with connected complements”, Journal of Combinatorial Theory, Series B 89 (2): 189–194, doi:10.1016/S0095-8956(03)00069-8, MR 2017723.

[16] Szegedy, Balázs; Szegedy, Christian (2006), “Symplectic spaces and ear-decomposition of matroids”, Combinatorica 26 (3): 353–377, doi:10.1007/s00493-006-0020-3, MR 2246153. Chapter 38

Forbidden graph characterization

In graph theory, a branch of mathematics, many important families of graphs can be described by a ﬁnite set of individual graphs that do not belong to the family and further exclude all graphs from the family which contain any of these forbidden graphs as (induced) subgraph or minor. A prototypical example of this phenomenon is Kuratowski’s theorem, which states that a graph is planar (can be drawn without crossings in the plane) if and only if it does not contain either of two forbidden graphs, the complete graph K5 and the complete bipartite graph K₃,₃. For Kuratowski’s theorem, the notion of containment is that of graph homeomorphism, in which a subdivision of one graph appears as a subgraph of the other. Thus, every graph either has a planar drawing (in which case it belongs to the family of planar graphs) or it has a subdivision of one of these two graphs as a subgraph (in which case it does not belong to the planar graphs). More generally, a forbidden graph characterization is a method of specifying a family of graph, or hypergraph, structures, by specifying substructures that are forbidden from existing within any graph in the family. Diﬀerent families vary in the nature of what is forbidden. In general, a structure G is a member of a family F if and only if a forbidden substructure is not contained in G. The forbidden substructure might be one of:

• subgraphs, smaller graphs obtained from subsets of the vertices and edges of a larger graph, • induced subgraphs, smaller graphs obtained by selecting a subset of the vertices and using all edges with both endpoints in that subset, • homeomorphic subgraphs (also called topological minors), smaller graphs obtained from subgraphs by collaps- ing paths of degree-two vertices to single edges, or • graph minors, smaller graphs obtained from subgraphs by arbitrary edge contractions.

The set of structures that are forbidden from belonging to a given graph family can also be called an obstruction set for that family. Forbidden graph characterizations may be used in algorithms for testing whether a graph belongs to a given family. In many cases, it is possible to test in polynomial time whether a given graph contains any of the members of the obstruction set, and therefore whether it belongs to the family defined by that obstruction set. In order for a family to have a forbidden graph characterization, with a particular type of substructure, the family must be closed under substructures. That is, every substructure (of a given type) of a graph in the family must be another graph in the family. Equivalently, if a graph is not part of the family, all larger graphs containing it as a substructure must also be excluded from the family. When this is true, there always exists an obstruction set (the set of graphs that are not in the family but whose smaller substructures all belong to the family). However, for some notions of what a substructure is, this obstruction set could be infinite. The Robertson–Seymour theorem proves that, for the particular case of graph minors, a family that is closed under minors always has a finite obstruction set.

38.1 List of forbidden characterizations for graphs and hypergraphs

This list is incomplete; you can help by expanding it.

141 142 CHAPTER 38. FORBIDDEN GRAPH CHARACTERIZATION

38.2 See also

• Erdős–Hajnal conjecture

• Forbidden subgraph problem

• Matroid minor

• Zarankiewicz problem

38.3 References

[1] Diestel, Reinhard (2000), Graph Theory, Graduate Texts in Mathematics 173, Springer-Verlag, ISBN 0-387-98976-5.

[2] Auer, Christopher; Bachmaier, Christian; Brandenburg, Franz J.; Gleißner, Andreas; Hanauer, Kathrin; Neuwirth, Daniel; Reislhuber, Josef (2013), “Recognizing outer 1-planar graphs in linear time”, in Wismath, Stephen; Wolﬀ, Alexander, 21st International Symposium, GD 2013, Bordeaux, France, September 23-25, 2013, Revised Selected Papers, Lecture Notes in Computer Science 8242, pp. 107–118, doi:10.1007/978-3-319-03841-4_10.

[3] Gupta, A.; Impagliazzo, R. (1991), “Computing planar intertwines”, Proc. 32nd IEEE Symposium on Foundations of Computer Science (FOCS '91), IEEE Computer Society, pp. 802–811, doi:10.1109/SFCS.1991.185452.

[4] Robertson, Neil; Seymour, P. D.; Thomas, Robin (1993), “Linkless embeddings of graphs in 3-space”, Bulletin of the Amer- ican Mathematical Society 28 (1): 84–89, arXiv:math/9301216, doi:10.1090/S0273-0979-1993-00335-5, MR 1164063.

[5] Béla Bollobás (1998) “Modern Graph Theory”, Springer, ISBN 0-387-98488-7 p. 9

[6] Kashiwabara, Toshinobu (1981), “Algorithms for some intersection graphs”, in Saito, Nobuji; Nishizeki, Takao, Graph Theory and Algorithms, 17th Symposium of Research Institute of Electric Communication, Tohoku University, Sendai, Japan, October 24-25, 1980, Proceedings, Lecture Notes in Computer Science 108, Springer-Verlag, pp. 171–181, doi:10.1007/3- 540-10704-5\_15.

[7] Chudnovsky, Maria; Robertson, Neil; Seymour, Paul; Thomas, Robin (2006), “The strong perfect graph theorem” (PDF), Annals of Mathematics 164 (1): 51–229, arXiv:math/0212070v1, doi:10.4007/annals.2006.164.51.

[8] Beineke, L. W. (1968), “Derived graphs of digraphs”, in Sachs, H.; Voss, H.-J.; Walter, H.-J., Beiträge zur Graphentheorie, Leipzig: Teubner, pp. 17–33.

[9] El-Mallah, Ehab; Colbourn, Charles J. (1988), “The complexity of some edge deletion problems”, IEEE Transactions on Circuits and Systems 35 (3): 354–362, doi:10.1109/31.1748.

[10] Takamizawa, K.; Nishizeki, Takao; Saito, Nobuji (1981), “Combinatorial problems on series-parallel graphs”, Discrete Applied Mathematics 3 (1): 75–76, doi:10.1016/0166-218X(81)90031-7.

[11] Joeris, Benson L.; Lin, Min Chih; McConnell, Ross M.; Spinrad, Jeremy P.; Szwarcﬁter, Jayme L. (2009), “Linear-Time Recognition of Helly Circular-Arc Models and Graphs”, Algorithmica 59 (2): 215–239, doi:10.1007/s00453-009-9304-5

[12] Földes, Stéphane; Hammer, Peter L. (1977a), “Split graphs”, Proceedings of the Eighth Southeastern Conference on Com- binatorics, Graph Theory and Computing (Louisiana State Univ., Baton Rouge, La., 1977), Congressus Numerantium XIX, Winnipeg: Utilitas Math., pp. 311–315, MR 0505860

[13] Bodlaender, Hans L. (1998), “A partial k-arboretum of graphs with bounded treewidth”, Theoretical Computer Science 209 (1–2): 1–45, doi:10.1016/S0304-3975(97)00228-4.

[14] Bodlaender, Hans L.; Thilikos, Dimitrios M. (1999), “Graphs with branchwidth at most three”, Journal of Algorithms 32 (2): 167–194, doi:10.1006/jagm.1999.1011.

[15] Seinsche, D. (1974), “On a property of the class of n-colorable graphs”, Journal of Combinatorial Theory, Series B 16 (2): 191–193, doi:10.1016/0095-8956(74)90063-X, MR 0337679

[16] Golumbic, Martin Charles (1978), “Trivially perfect graphs”, Discrete Mathematics 24 (1): 105–107, doi:10.1016/0012- 365X(78)90178-4..

[17] Metelsky, Yury; Tyshkevich, Regina (1997), “On line graphs of linear 3-uniform hypergraphs”, Journal of Graph Theory 25 (4): 243–251, doi:10.1002/(SICI)1097-0118(199708)25:4<243::AID-JGT1>3.0.CO;2-K, MR 1459889 38.3. REFERENCES 143

[18] Jacobson, M. S.; Kézdy, Andre E.; Lehel, Jeno (1997), “Recognizing intersection graphs of linear uniform hypergraphs”, Graphs and Combinatorics 13: 359–367, doi:10.1007/BF03353014, MR 1485929

[19] Naik, Ranjan N.; Rao, S. B.; Shrikhande, S. S.; Singhi, N. M. (1982), “Intersection graphs of k-uniform hypergraphs”, European J. Combinatorics 3: 159–172, doi:10.1016/s0195-6698(82)80029-2, MR 0670849 Chapter 39

Gallery of named graphs

Some of the ﬁnite structures considered in graph theory have names, sometimes inspired by the graph’s topology, and sometimes after their discoverer. A famous example is the Petersen graph, a concrete graph on 10 vertices that appears as a minimal example or counterexample in many diﬀerent contexts.

39.1 Individual graphs

• Balaban 10-cage

• Balaban 11-cage

• Bidiakis cube

• Brinkmann graph

• Bull graph

• Butterﬂy graph

• Chvátal graph

• Diamond graph

• Dürer graph

• Ellingham–Horton 54-graph

• Ellingham–Horton 78-graph

• Errera graph

• Franklin graph

• Frucht graph

• Goldner–Harary graph

• Grötzsch graph

• Harries graph

• Harries–Wong graph

• Herschel graph

• Hoﬀman graph

• Holt graph

144 39.2. HIGHLY SYMMETRIC GRAPHS 145

• Horton graph

• Kittell graph

• Markström graph

• McGee graph

• Meredith graph

• Moser spindle

• Sousselier graph

• Poussin graph

• Robertson graph

• Sylvester graph

• Tutte’s fragment

• Tutte graph

• Young–Fibonacci graph

• Wagner graph

• Wells graph

• Wiener–Araya graph

39.2 Highly symmetric graphs

39.2.1 Strongly regular graphs

The strongly regular graph on v vertices and rank k is usually denoted srg(v,k,λ,μ).

• Clebsch graph

• Cameron graph

• Petersen graph

• Hall–Janko graph

• Hoﬀman–Singleton graph

• Higman–Sims graph

• Paley graph of order 13

• Shrikhande graph

• Schläﬂi graph

• Brouwer–Haemers graph

• Local McLaughlin graph

• Perkel graph

• Gewirtz graph 146 CHAPTER 39. GALLERY OF NAMED GRAPHS

39.2.2 Symmetric graphs

A symmetric graph is one in which there is a symmetry (graph automorphism) taking any ordered pair of adjacent vertices to any other ordered pair; the Foster census lists all small symmetric 3-regular graphs. Every strongly regular graph is symmetric, but not vice versa.

• Heawood graph

• Möbius–Kantor graph

• Pappus graph

• Desargues graph

• Nauru graph

• Coxeter graph

• Tutte–Coxeter graph

• Dyck graph

• Klein graph

• Foster graph

• Biggs–Smith graph

• The Rado graph

39.2.3 Semi-symmetric graphs

• Folkman graph

• Gray graph

• Ljubljana graph

• Tutte 12-cage

39.3 Graph families

39.3.1 Complete graphs

[1] The complete graph on n vertices is often called the n -clique and usually denoted Kn , from German komplett.

•

• 39.3. GRAPH FAMILIES 147

39.3.2 Complete bipartite graphs

The complete bipartite graph is usually denoted Kn,m . For n = 1 see the section on star graphs. The graph K2,2 equals the 4-cycle C4 (the square) introduced below.

•

• , the utility graph •

•

39.3.3 Cycles

The cycle graph on n vertices is called the n-cycle and usually denoted Cn . It is also called a cyclic graph, a polygon or the n-gon. Special cases are the triangle C3 , the square C4 , and then several with Greek naming pentagon C5 , hexagon C6 , etc.

•

39.3.4 Friendship graphs

[2] The friendship graph Fn can be constructed by joining n copies of the cycle graph C3 with a common vertex.

The friendship graphs F2, F3 and F4.

39.3.5 Fullerene graphs

In graph theory, the term fullerene refers to any 3-regular, planar graph with all faces of size 5 or 6 (including the external face). It follows from Euler’s polyhedron formula, V – E + F = 2 (where V, E, F indicate the number of vertices, edges, and faces), that there are exactly 12 pentagons in a fullerene and V/2–10 hexagons. Fullerene graphs are the Schlegel representations of the corresponding fullerene compounds.

• 20-fullerene (dodecahedral graph)

• 24-fullerene (Hexagonal truncated trapezohedron graph)

• 26-fullerene 148 CHAPTER 39. GALLERY OF NAMED GRAPHS

• 60-fullerene (truncated icosahedral graph)

• 70-fullerene

An algorithm to generate all the non-isomorphic fullerens with a given number of hexagonal faces has been developed by G. Brinkmann and A. Dress.[3] G. Brinkmann also provided a freely available implementation, called fullgen.

39.3.6 Platonic solids

The complete graph on four vertices forms the skeleton of the tetrahedron, and more generally the complete graphs form skeletons of simplices. The hypercube graphs are also skeletons of higher-dimensional regular polytopes.

• Cube ,

• Octahedron ,

• Dodecahedron ,

• Icosahedron ,

39.3.7 Truncated solids

• Truncated tetrahedron

• Truncated cube

• Truncated octahedron

• Truncated dodecahedron

• Truncated icosahedron

39.3.8 Snarks

A snark is a bridgeless cubic graph that requires four colors in any proper edge coloring. The smallest snark is the Petersen graph, already listed above.

• Blanuša snark (ﬁrst)

• Blanuša snark (second)

• Double-star snark

• Flower snark

• Loupekine snark (ﬁrst)

• Loupekine snark (second)

• Szekeres snark

• Tietze graph

• Watkins snark 39.4. REFERENCES 149

The star graphs S3, S4, S5 and S6.

39.3.9 Star

A star S is the complete bipartite graph K₁,k. The star S3 is called the claw graph.

39.3.10 Wheel graphs

The wheel graph Wn is a graph on n vertices constructed by connecting a single vertex to every vertex in an (n − 1)-cycle.

Wheels W4 – W9 .

39.4 References

[1] David Gries and Fred B. Schneider, A Logical Approach to Discrete Math, Springer, 1993, p 436. [2] Gallian, J. A. “Dynamic Survey DS6: Graph Labeling.” Electronic Journal of Combinatorics, DS6, 1-58, January 3, 2007. . [3] Journal of Algorithms 23 (2): 345–358. 1997. doi:10.1006/jagm.1996.0806. MR 1441972. Missing or empty |title= (help) Chapter 40

Half-transitive graph

In the mathematical ﬁeld of graph theory, a half-transitive graph is a graph that is both vertex-transitive and edge- transitive, but not symmetric.[1] In other words, a graph is half-transitive if its automorphism group acts transitively upon both its vertices and its edges, but not on ordered pairs of linked vertices. Every connected symmetric graph must be vertex-transitive and edge-transitive, and the converse is true for graphs of odd degree,[2] so that half-transitive graphs of odd degree do not exist. However, there do exist half-transitive graphs of even degree.[3] The smallest half-transitive graph is the Holt graph, with degree 4 and 27 vertices.[4][5]

40.1 References

[1] Gross, J.L. and Yellen, J. (2004). Handbook of Graph Theory. CRC Press. p. 491. ISBN 1-58488-090-2.

[2] Babai, L (1996). “Automorphism groups, isomorphism, reconstruction”. In Graham, R; Grötschel, M; Lovász, L. Hand- book of Combinatorics. Elsevier.

[3] Bouwer, Z. “Vertex and Edge Transitive, But Not 1-Transitive Graphs.” Canad. Math. Bull. 13, 231–237, 1970.

[4] Biggs, Norman (1993). Algebraic Graph Theory (2nd ed.). Cambridge: Cambridge University Press. ISBN 0-521-45897- 8.

[5] Holt, Derek F. (1981). “A graph which is edge transitive but not arc transitive”. Journal of Graph Theory 5 (2): 201–204. doi:10.1002/jgt.3190050210..

150 40.1. REFERENCES 151

The Holt graph is the smallest half-transitive graph. The lack of reﬂectional symmetry in this drawing highlights the fact that edges are not equivalent to their inverse. Chapter 41

Halin graph

A Halin graph.

In graph theory, a Halin graph is a type of planar graph. It is constructed from a tree that has at least four vertices, none of which have exactly two neighbors. The tree is drawn in the plane so none of its edges cross; then edges are added that connect all its leaves into a cycle.[1] Halin graphs are named after German mathematician Rudolf Halin, who studied them in 1971,[2] but the cubic Halin graphs had already been studied over a century earlier by Kirkman.[3]

152 41.1. CONSTRUCTION 153

A triangular prism, constructed as a Halin graph from a six-vertex tree

41.1 Construction

A Halin graph is constructed as follows. Let T be a tree with more than three vertices, such that no vertex of T has degree two (that is, no vertex has exactly two neighbors), embedded in the plane. Then a Halin graph is constructed by adding to T a cycle through each of its leaves, such that the augmented graph remains planar.

41.2 Examples

A star is a tree with exactly one internal vertex. Applying the Halin graph construction to a star produces a wheel graph, the graph of a pyramid. The graph of a triangular prism is also a Halin graph: it can be drawn so that one of its rectangular faces is the exterior cycle, and the remaining edges form a tree with four leaves, two interior vertices, and ﬁve edges. The Frucht graph, one of the two smallest cubic graphs with no nontrivial graph automorphisms, is also a Halin graph.

41.3 Properties

Every Halin graph is 3-connected, meaning that it is not possible to delete two vertices from it and disconnect the remaining vertices. It is edge-minimal 3-connected, meaning that if any one of its edges is removed, the remaining graph will no longer be 3-connected.[1] By Steinitz’s theorem, as a 3-connected planar graph, it can be represented as the set of vertices and edges of a convex polyhedron; that is, it is a polyhedral graph. And, as with every polyhedral graph, its planar embedding is unique up to the choice of which of its faces is to be the outer face.[1] Every Halin graph is a Hamiltonian graph, and every edge of the graph belongs to a Hamiltonian cycle. Moreover, any Halin graph remains Hamiltonian after deletion of any vertex.[4] Because every tree without vertices of degree 2 contains two leaves that share the same parent, every Halin graph contains a triangle. In particular, it is not possible 154 CHAPTER 41. HALIN GRAPH

Wheel graphs

for a Halin graph to be a triangle-free graph nor a bipartite graph. More strongly, every Halin graph is almost pancyclic, in the sense that it has cycles of all lengths from 3 to n with the possible exception of a single even length. Moreover, any Halin graph remains almost pancyclic if a single edge is contracted, and every Halin graph without interior vertices of degree three is pancyclic.[5] Every Halin graph has treewidth at most three.[6] Therefore, many graph optimization problems that are NP-complete for arbitrary planar graphs, such as ﬁnding a maximum independent set, may be solved in linear time on Halin graphs using dynamic programming.[7] The weak dual of an embedded planar graph has vertices corresponding to bounded faces of the planar graph, and edges corresponding to adjacent faces. The weak dual of a Halin graph is always biconnected and outerplanar. This property may be used to characterize the Halin graphs: an embedded planar graph is a Halin graph, with the leaf cycle of the Halin graph as the outer face of the embedding, if and only if its weak dual is biconnected and outerplanar.[8] The incidence chromatic number of a Halin graph G with maximum degree Δ(G) greater than four is Δ(G) + 1.[9] When Δ(G) = 3 or 4, the incidence chromatic number may be as large as 5 or 6 respectively.[10]

41.4 Computational complexity

It is possible to test whether a given n-vertex graph is a Halin graph in linear time, by ﬁnding a planar embedding of the graph (if one exists), and then testing whether there exists a face that has at least n/2 + 1 vertices, all of degree three. If so, there can be at most four such faces, and it is possible to check in linear time for each of them whether the rest of the graph forms a tree with the vertices of this face as its leaves. On the other hand, if no such face exists, then the graph is not Halin.[11] Alternatively, a graph with n vertices and m edges is Halin if and only if it is planar, 3-connected, and has a face whose number of vertices equals the circuit rank m − n + 1 of the graph, all of which can be checked in linear time.[8] However, it is NP-complete to ﬁnd the largest Halin subgraph of a given graph, to test whether there exists a Halin subgraph that includes all vertices of a given graph, or to test whether a given graph is a subgraph of a larger Halin graph.[12] 41.5. HISTORY 155

41.5 History

In 1971, Halin introduced the Halin graphs as a class of minimally 3-vertex-connected graphs: for every edge in the graph, the removal of that edge reduces the connectivity of the graph.[2] These graphs gained in significance with the discovery that many algorithmic problems that were computationally infeasible for arbitrary planar graphs could be solved efficiently on them.[4][8] This fact was later explained to be a consequence of their low treewidth, and of algorithmic meta-theorems like Courcelle’s theorem that provide efficient solutions to these problems on any graph of low treewidth.[6][7] Prior to Halin’s work on these graphs, graph enumeration problems concerning the cubic Halin graphs were studied in 1856 by Thomas Kirkman[3] and in 1965 by Hans Rademacher.[13] Rademacher calls these graphs based polyhedra. He defines them as the cubic polyhedral graphs with f faces in which one of the faces has f − 1 sides. The graphs that fit this definition are exactly the cubic Halin graphs. The Halin graphs are sometimes also called roofless polyhedra,[4] but, like “based polyhedra”, this name may also refer to the cubic Halin graphs.[14] The convex polyhedra whose graphs are Halin graphs have also been called domes.[15]

41.6 References

[1] Encyclopaedia of Mathematics, ﬁrst Supplementary volume, 1988, ISBN 0-7923-4709-9, p. 281, article “Halin Graph”, and references therein. [2] Halin, R. (1971), “Studies on minimally n-connected graphs”, Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969), London: Academic Press, pp. 129–136, MR 0278980. [3] Kirkman, Th. P. (1856), “On the enumeration of x-edra having triedral summits and an (x − 1)-gonal base”, Philosophical Transactions of the Royal Society of London: 399–411, doi:10.1098/rstl.1856.0018, JSTOR 108592. [4] Cornuéjols, G.; Naddef, D.; Pulleyblank, W. R. (1983), “Halin graphs and the travelling salesman problem”, Mathematical Programming 26 (3): 287–294, doi:10.1007/BF02591867. [5] Skowrońska, Mirosława (1985), “The pancyclicity of Halin graphs and their exterior contractions”, in Alspach, Brian R.; Godsil, Christopher D., Cycles in Graphs, Annals of Discrete Mathematics 27, Elsevier Science Publishers B.V., pp. 179– 194. [6] Bodlaender, Hans (1988), Planar graphs with bounded treewidth (PDF), Technical Report RUU-CS-88-14, Department of Computer Science, Utrecht University. [7] Bodlaender, Hans (1988), “Dynamic programming on graphs with bounded treewidth”, Proceedings of the 15th Interna- tional Colloquium on Automata, Languages and Programming, Lecture Notes in Computer Science 317, Springer-Verlag, pp. 105–118, doi:10.1007/3-540-19488-6_110, ISBN 978-3540194880. [8] Sysło, Maciej M.; Proskurowski, Andrzej (1983), “On Halin graphs”, Graph Theory: Proceedings of a Conference held in Lagów, Poland, February 10–13, 1981, Lecture Notes in Mathematics 1018, Springer-Verlag, pp. 248–256, doi:10.1007/BFb0071635. [9] Wang, Shu-Dong; Chen, Dong-Ling; Pang, Shan-Chen (2002), “The incidence coloring number of Halin graphs and outerplanar graphs”, Discrete Mathematics 256 (1-2): 397–405, doi:10.1016/S0012-365X(01)00302-8, MR 1927561. [10] Shiu, W. C.; Sun, P. K. (2008), “Invalid proofs on incidence coloring”, Discrete Mathematics 308 (24): 6575–6580, doi:10.1016/j.disc.2007.11.030, MR 2466963. [11] Fomin, Fedor V.; Thilikos, Dimitrios M. (2006), “A 3-approximation for the pathwidth of Halin graphs”, Journal of Discrete Algorithms 4 (4): 499–510, doi:10.1016/j.jda.2005.06.004, MR 2577677. [12] Horton, S. B.; Parker, R. Gary (1995), “On Halin subgraphs and supergraphs”, Discrete Applied Mathematics 56 (1): 19–35, doi:10.1016/0166-218X(93)E0131-H, MR 1311302. [13] Rademacher, Hans (1965), “On the number of certain types of polyhedra”, Illinois Journal of Mathematics 9: 361–380, MR 0179682. [14] Lovász, L.; Plummer, M. D. (1974), “On a family of planar bicritical graphs”, Combinatorics (Proc. British Combinatorial Conf., Univ. Coll. Wales, Aberystwyth, 1973), London: Cambridge Univ. Press, pp. 103–107. London Math. Soc. Lecture Note Ser., No. 13, MR 0351915. [15] Demaine, Erik D.; Demaine, Martin L.; Uehara, Ryuhei (2013), “Zipper unfolding of domes and prismoids”, Proceedings of the 25th Canadian Conference on Computational Geometry (CCCG 2013), Waterloo, Ontario, Canada, August 8–10, 2013, pp. 43–48. 156 CHAPTER 41. HALIN GRAPH

41.7 External links

• Halin graphs, Information System on Graph Class Inclusions.

• Weisstein, Eric W., “Halin Graph”, MathWorld. Chapter 42

Hanan grid

Hanan grid generated for a 5-terminal case

In geometry, the Hanan grid H(S) of a ﬁnite set S of points in the plane is obtained by constructing vertical and

157 158 CHAPTER 42. HANAN GRID horizontal lines through each point in S. The main motivation for studying the Hanan grid stems from the fact that it is known to contain a rectilinear Steiner tree for S.[1] It is named after Maurice Hanan, who was ﬁrst[2] to investigate the rectilinear Steiner minimum tree and introduced this graph.[3]

42.1 References

[1] Martin Zachariasen, A Catalog of Hanan Grid Problems Networks, vol. 38, 2000, pp. 200-221

[2] Christine R. Leverenz, Miroslaw Truszczynski, The Rectilinear Steiner Tree Problem: Algorithms and Examples using Permutations of the Terminal Set, 1999 ACM Southeast Regional Conference, 1999, doi:10.1145/306363.306402

[3] M. Hanan, On Steiner’s problem with rectilinear distance, J. SIAM Appl. Math. 14 (1966), 255 - 265. Chapter 43

Highly irregular graph

In graph theory, a highly irregular graph is a graph in which, for every vertex, all neighbors of that vertex have distinct degrees.

43.1 History

Irregular graphs were initially characterized by Yousef Alavi, Gary Chartrand, Fan Chung, Paul Erdős, R L Graham, and Ortrud Oellerman.[1] They were motivated to deﬁne the ‘opposite’ of a regular graph, a concept which has been thoroughly studied and well-understood.

43.2 Locality and regularity

Defining an ‘irregular graph’ was not immediately obvious. In a k-regular graph, all vertices have degree k. In any graph G, two vertices in G must have the same degree, so an irregular graph cannot be defined as a graph with all vertices of different degrees. One may be tempted then to define an irregular graph as having all vertices of distinct degrees except for two, but these types of graphs are also well understood and thus not interesting.[2] Graph theorists thus turned to the issue of local regularity. A graph is locally regular at a vertex v if all vertices adjacent to v have degree r. A graph is thus locally irregular if for each vertex v of G the neighbors of v have distinct degrees, and these graphs are thus termed highly irregular graphs.[1]

43.3 Properties of irregular graphs

Some facts about highly irregular graphs outlined by Alavi et al.:[3]

• If v is a vertex of maximum degree d in a highly irregular graph H, then v is adjacent to exactly one vertex of degree 1, 2, ..., d.[3]

• The largest degree in a highly irregular graph is at most half the number of vertices.[3]

• If H is a highly irregular graph with maximum degree d, one can construct a highly irregular graph of degree d+1 by taking two copies of H and adding an edge between the two vertices of degree d.[3]

• H(n)/G(n) goes to 0 as n goes to inﬁnity exponentially rapidly, where H(n) is the number of (non-isomorphic) highly irregular graphs with n vertices, and G(n) is the total number of graphs with n vertices.[3]

• For every graph G, there exists a highly irregular graph H containing G as an induced subgraph.[3]

This last observation can be considered analogous to a result of Dénes Kőnig, which states that if H is a graph with greatest degree r, then there is a graph G which is r-regular and contains H as an induced subgraph.[3]

159 160 CHAPTER 43. HIGHLY IRREGULAR GRAPH

43.4 Applications of irregularity

Definitions of irregularity have been important in the study of network heterogeneity, which has implications in networks found across biology, ecology, technology, and economy.[4] There have been several graph statistics that have been suggested, many of which are based on the number of vertices in a graph and their degrees. The characterization of highly irregular graphs has also been applied to the question of heterogeneity, yet all of these fail to shed enough light on real-world situations. Efforts continue to be made to find appropriate ways to quantify network heterogeneity.[4]

43.5 References

[1] Chartrand, Gary, Paul Erdos, and Ortrud R. Oellermann. “How to deﬁne an irregular graph.” College Math. J 19.1 (1988): 36–42.

[2] Behzad, Mehdi, and Gary Chartrand. “No graph is perfect.” American Mathematical Monthly (1967): 962–963.

[3] Alavi, Yousef, et al. “Highly irregular graphs.” Journal of graph theory 11.2 (1987): 235–249.

[4] Estrada, Ernesto. “Quantifying network heterogeneity.” Physical Review E 82.6 (2010): 066102. Chapter 44

Homogeneous graph

In mathematics, a k-ultrahomogeneous graph is a graph in which every isomorphism between two of its induced subgraphs of at most k vertices can be extended to an automorphism of the whole graph. A a k-homogeneous graph obeys a weakened version of the same property in which every isomorphism between two induced subgraphs implies the existence of an automorphism of the whole graph that maps one subgraph to the other (but does not necessarily extend the given isomorphism).[1] A homogeneous graph is a graph that is k-homogeneous for every k, or equivalently k-ultrahomogeneous for every k.[1]

44.1 Classiﬁcation

The only finite homogeneous graphs are the graphs mKn formed from the disjoint unions of isomorphic complete graphs, the Turán graphs formed as the complement graphs of mKn, the 3 × 3 rook’s graphs, and the 5-cycle.[2] The only countably infinite homogeneous graphs are the disjoint unions of isomorphic complete graphs (with the size of each complete graph, the number of complete graphs, or both numbers countably infinite), their complement graphs, the Rado graph, and the Henson graphs.[3] If a graph is 5-ultrahomogeneous, then it is ultrahomogeneous for every k. There are only two connected graphs that are 4-ultrahomogeneous but not 5-ultrahomogeneous: the Schläfli graph and its complement. The proof relies on the classification of finite simple groups.[4]

44.2 Notes

[1] Ronse (1978).

[2] Gardiner (1976).

[3] Lachlan & Woodrow (1980).

[4] Buczak (1980); Cameron (1980); Devillers (2002).

44.3 References

• Buczak, J. M. J. (1980), Finite Group Theory, Ph.D. thesis, Oxford University. As cited by Devillers (2002).

• Cameron, Peter Jephson (1980), “6-transitive graphs”, Journal of Combinatorial Theory, Series B 28: 168– 179, doi:10.1016/0095-8956(80)90063-5. As cited by Devillers (2002).

• Devillers, Alice (2002), Classiﬁcation of some homogeneous and ultrahomogeneous structures, Ph.D. thesis, Université Libre de Bruxelles.

161 162 CHAPTER 44. HOMOGENEOUS GRAPH

• Gardiner, A. (1976), “Homogeneous graphs”, Journal of Combinatorial Theory, Series B 20 (1): 94–102, MR 0419293. • Lachlan, A. H.; Woodrow, Robert E. (1980), “Countable ultrahomogeneous undirected graphs”, Transactions of the American Mathematical Society 262 (1): 51–94, doi:10.2307/1999974, MR 583847. • Ronse, Christian (1978), “On homogeneous graphs”, Journal of the London Mathematical Society, Second Series 17 (3): 375–379, doi:10.1112/jlms/s2-17.3.375, MR 500619. Chapter 45

Hypohamiltonian graph

Not to be confused with Hamiltonian graph. In the mathematical ﬁeld of graph theory, a graph G is said to be hypohamiltonian if G does not itself have a

A hypohamiltonian graph constructed by Lindgren (1967).

Hamiltonian cycle but every graph formed by removing a single vertex from G is Hamiltonian.

163 164 CHAPTER 45. HYPOHAMILTONIAN GRAPH

45.1 History

Hypohamiltonian graphs were ﬁrst studied by Sousselier (1963). Lindgren (1967) cites Gaudin, Herz & Rossi (1964) and Busacker & Saaty (1965) as additional early papers on the subject; another early work is by Herz, Duby & Vigué (1967). Grötschel (1980) sums up much of the research in this area with the following sentence: “The articles dealing with those graphs ... usually exhibit new classes of hypohamiltonian or hypotraceable graphs showing that for certain orders n such graphs indeed exist or that they possess strange and unexpected properties.”

45.2 Applications

Hypohamiltonian graphs arise in integer programming solutions to the traveling salesman problem: certain kinds of hypohamiltonian graphs define facets of the traveling salesman polytope, a shape defined as the convex hull of the set of possible solutions to the traveling salesman problem, and these facets may be used in cutting-plane methods for solving the problem.[1] Grötschel (1980) observes that the computational complexity of determining whether a graph is hypohamiltonian, although unknown, is likely to be high, making it difficult to find facets of these types except for those defined by small hypohamiltonian graphs; fortunately, the smallest graphs lead to the strongest inequalities for this application.[2] Concepts closely related to hypohamiltonicity have also been used by Park, Lim & Kim (2007) to measure the fault tolerance of network topologies for parallel computing.

45.3 Properties

Every hypohamiltonian graph must be 3-vertex-connected, as the removal of any two vertices leaves a Hamiltonian path, which is connected. There exist n-vertex hypohamiltonian graphs in which the maximum degree is n/2, and in which there are approximately n2/4 edges.[3] Herz, Duby & Vigué (1967) conjectured that every hypohamiltonian graph has girth 5 or more, but this was disproved by Thomassen (1974b), who found examples with girth 3 and 4. For some time it was unknown whether a hypohamiltonian graph could be planar, but several examples are now known,[4] the smallest of which has 40 vertices.[5] Every planar hypohamiltonian graph has at least one vertex with only three incident edges.[6] If a 3-regular graph is Hamiltonian, its edges can be colored with three colors: use alternating colors for the edges on the Hamiltonian cycle (which must have even length by the handshaking lemma) and a third color for all remaining edges. Therefore, all snarks, bridgeless cubic graphs that require four edge colors, must be non-Hamiltonian, and many known snarks are hypohamiltonian. Every hypohamiltonian snark is bicritical: removing any two vertices leaves a subgraph the edges of which can be colored with only three colors.[7] A three-coloring of this subgraph can be simply described: after removing one vertex, the remaining vertices contain a Hamiltonian cycle. After removing a second vertex, this cycle becomes a path, the edges of which may be colored by alternating between two colors. The remaining edges form a matching and may be colored with a third color. The color classes of any 3-coloring of the edges of a 3-regular graph form three matchings such that each edge belongs to exactly one of the matchings. Hypohamiltonian snarks do not have a partition into matchings of this type, but Häggkvist (2007) conjectures that the edges of any hypohamiltonian snark may be used to form six matchings such that each edge belongs to exactly two of the matchings. This is a special case of the Berge–Fulkerson conjecture that any snark has six matchings with this property. Hypohamiltonian graphs cannot be bipartite: in a bipartite graph, a vertex can only be deleted to form a Hamiltonian subgraph if it belongs to the larger of the graph’s two color classes. However, every bipartite graph occurs as an induced subgraph of some hypohamiltonian graph.[8]

45.4 Examples

The smallest hypohamiltonian graph is the Petersen graph (Herz, Duby & Vigué 1967). More generally, the generalized Petersen graph GP(n,2) is hypohamiltonian when n is 5 (mod 6);[9] the Petersen graph is the instance of this con- 45.5. ENUMERATION 165

Thomassen’s (1974b) girth-3 hypohamiltonian graph.

struction with n = 5. Lindgren (1967) found another inﬁnite class of hypohamiltonian graphs in which the number of vertices is 4 (mod 6). Lindgren’s construction consists of a cycle of length 3 (mod 6) and a single central vertex; the central vertex is connected to every third vertex of the cycle by edges he calls spokes, and the vertices two positions away from each spoke endpoint are connected to each other. Again, the smallest instance of Lindgren’s construction is the Petersen graph. Additional inﬁnite families are given by Bondy (1972), Doyen & van Diest (1975), and Gutt (1977).

45.5 Enumeration

Václav Chvátal proved in 1973 that for all sufficiently large n there exists a hypohamiltonian graph with n vertices. Taking into account subsequent discoveries,[10] “sufficiently large” is now known to mean that such graphs exist for all n ≥ 18. A complete list of hypohamiltonian graphs with at most 17 vertices is known:[11] they are the 10-vertex Petersen graph, a 13-vertex graph and a 15-vertex graph found by computer searches of Herz (1968), and four 16- vertex graphs. There exist at least thirteen 18-vertex hypohamiltonian graphs. By applying the flip-flop method of Chvátal (1973) to the Petersen graph and the flower snark, it is possible to show that the number of hypohamiltonian graphs, and more specifically the number of hypohamiltonian snarks, grows as an exponential function of the number of vertices.[12] 166 CHAPTER 45. HYPOHAMILTONIAN GRAPH

45.6 Generalizations

Graph theorists have also studied hypotraceable graphs, graphs that do not contain a Hamiltonian path but such that every subset of n − 1 vertices may be connected by a path.[13] Analogous definitions of hypohamiltonicity and hypotraceability for directed graphs have been considered by several authors.[14] An equivalent definition of hypohamiltonian graphs is that their longest cycle has length n − 1 and that the intersection of all longest cycles is empty. Menke, Zamfirescu & Zamfirescu (1998) investigate graphs with the same property that the intersection of longest cycles is empty but in which the longest cycle length is shorter than n − 1. Herz (1968) defines the cyclability of a graph as the largest number k such that every k vertices belong to a cycle; the hypohamiltonian graphs are exactly the graphs that have cyclability n − 1. Similarly, Park, Lim & Kim (2007) define a graph to be ƒ-fault hamiltonian if the removal of at most ƒ vertices leaves a Hamiltonian subgraph. Schauerte & Zamfirescu (2006) study the graphs with cyclability n − 2.

45.7 Notes

[1] Grötschel (1977); Grötschel (1980); Grötschel & Wakabayashi (1981).

[2] Goemans (1995).

[3] Thomassen (1981).

[4] The existence of planar hypohamiltonian graphs was posed as an open question by Chvátal (1973), and Chvátal, Klarner & Knuth (1972) offered a $5 prize for the construction of one. Thomassen (1976) used Grinberg’s theorem to find planar hypohamiltonian graphs of girth 3, 4, and 5 and showed that there exist infinitely many planar hypohamiltonian graphs.

[5] Jooyandeh et al. (2013), using a computer search and Grinberg’s theorem. Earlier small planar hypohamiltonian graphs with 42, 57 and 48 vertices, respectively, were found by Wiener & Araya (2009), Hatzel (1979) and Zamﬁrescu & Zamﬁrescu (2007).

[6] Thomassen (1978).

[7] Steﬀen (1998); Steﬀen (2001).

[8] Collier & Schmeichel (1977).

[9] Robertson (1969) proved that these graphs are non-Hamiltonian, while it is straightforward to verify that their one-vertex deletions are Hamiltonian. See Alspach (1983) for a classiﬁciation of non-Hamiltonian generalized Petersen graphs.

[10] Thomassen (1974a); Doyen & van Diest (1975).

[11] Aldred, McKay & Wormald (1997). See also (sequence A141150 in OEIS).

[12] Skupień (1989); Skupień (2007).

[13] Kapoor, Kronk & Lick (1968); Kronk (1969); Grünbaum (1974); Thomassen (1974a).

[14] Fouquet & Jolivet (1978); Grötschel & Wakabayashi (1980); Grötschel & Wakabayashi (1984); Thomassen (1978).

45.8 References

• Aldred, R. A.; McKay, B. D.; Wormald, N. C. (1997), “Small hypohamiltonian graphs” (PDF), J. Combina- torial Math. Combinatorial Comput. 23: 143–152. • Alspach, B. R. (1983), “The classiﬁcation of Hamiltonian generalized Petersen graphs”, Journal of Combina- torial Theory, Series B 34 (3): 293–312, doi:10.1016/0095-8956(83)90042-4, MR 0714452. • Bondy, J. A. (1972), “Variations of the Hamiltonian theme”, Canadian Mathematical Bulletin 15: 57–62, doi:10.4153/CMB-1972-012-3. • Busacker, R. G.; Saaty, T. L. (1965), Finite Graphs and Networks, McGraw-Hill. • Chvátal, V. (1973), “Flip-ﬂops in hypo-Hamiltonian graphs”, Canadian Mathematical Bulletin 16: 33–41, doi:10.4153/CMB-1973-008-9, MR 0371722. 45.8. REFERENCES 167

• Chvátal, V.; Klarner, D. A.; Knuth, D. E. (1972), Selected Combinatorial Research Problems (PDF), Tech. Report STAN-CS-72-292, Computer Science Department, Stanford University.

• Collier, J. B.; Schmeichel, E. F. (1977), “New ﬂip-ﬂop constructions for hypohamiltonian graphs”, Discrete Mathematics 18 (3): 265–271, doi:10.1016/0012-365X(77)90130-3, MR 0543828.

• Doyen, J.; van Diest, V. (1975), “New families of hypohamiltonian graphs”, Discrete Mathematics 13 (3): 225–236, doi:10.1016/0012-365X(75)90020-5, MR 0416979.

• Fouquet, J.-L.; Jolivet, J. L. (1978), “Hypohamiltonian oriented graphs”, Cahiers Centre Études Rech. Opér. 20 (2): 171–181, MR 0498218.

• Gaudin, T.; Herz, P.; Rossi (1964), “Solution du problème No. 29”, Rev. Franç. Rech. Opérationnelle 8: 214–218.

• Goemans, Michel X. (1995), “Worst-case comparison of valid inequalities for the TSP”, Mathematical Pro- gramming 69 (1–3): 335–349, doi:10.1007/BF01585563.

• Grötschel, M. (1977), “Hypohamiltonian facets of the symmetric travelling salesman polytope”, Zeitschrift für Angewandte Mathematik und Mechanik 58: 469–471.

• Grötschel, M. (1980), “On the monotone symmetric traveling salesman problem: hypohamiltonian/hypotraceable graphs and facets”, Mathematics of Operations Research 5 (2): 285–292, doi:10.1287/moor.5.2.285, JSTOR 3689157.

• Grötschel, M.; Wakabayashi, Y. (1980), “Hypohamiltonian digraphs”, Mathematics of Operations Research 36: 99–119.

• Grötschel, M.; Wakabayashi, Y. (1981), “On the structure of the monotone asymmetric travelling salesman polytope I: hypohamiltonian facets”, Discrete Mathematics 24: 43–59, doi:10.1016/0012-365X(81)90021-2.

• Grötschel, M.; Wakabayashi, Y. (1984), “Constructions of hypotraceable digraphs”, in Cottle, R. W.; Kel- manson, M. L.; Korte, B., Mathematical Programming (Proc. International Congress, Rio de Janeiro, 1981), North-Holland.

• Grünbaum, B. (1974), “Vertices missed by longest paths or circuits”, Journal of Combinatorial Theory, Series A 17: 31–38, doi:10.1016/0097-3165(74)90025-9, MR 0349474.

• Gutt, S. (1977), “Inﬁnite families of hypohamiltonian graphs”, Académie Royale de Belgique, Bulletin de la Classe des Sciences, Koninklijke Belgische Academie, Mededelingen van de Klasse der Wetenschappen, 5e Série 63 (5): 432–440, MR 0498243.

• Häggkvist, R. (2007), “Problem 443. Special case of the Fulkerson Conjecture”, in Mohar, B.; Nowakowski, R. J.; West, D. B., Research problems from the 5th Slovenian Conference (Bled, 2003), Discrete Mathematics 307 (3–5): 650–658, doi:10.1016/j.disc.2006.07.013.

• Hatzel, W. (1979), “Ein planarer hypohamiltonscher Graph mit 57 Knoten”, Math. Ann. 243 (3): 213–216, doi:10.1007/BF01424541, MR 0548801.

• Herz, J. C. (1968), “Sur la cyclabilité des graphes”, Computers in Mathematical Research, North-Holland, pp. 97–107, MR 0245461.

• Herz, J. C.; Duby, J. J.; Vigué, F. (1967), “Recherche systématique des graphes hypohamiltoniens”, in Rosenstiehl, P., Theory of Graphs: International Symposium, Rome 1966, Paris: Gordon and Breach, pp. 153–159.

• Jooyandeh, Mohammadreza; McKay, Brendan D.; Östergård, Patric R. J.; Pettersson, Ville H.; Zamﬁrescu, Carol T. (2013), Planar Hypohamiltonian Graphs on 40 Vertices, arXiv:1302.2698.

• Kapoor, S. F.; Kronk, H. V.; Lick, D. R. (1968), “On detours in graphs”, Canadian Mathematical Bulletin 11: 195–201, doi:10.4153/CMB-1968-022-8, MR 0229543.

• Kronk, H. V. (1969), Klee, V., ed., “Does there exist a hypotraceable graph?", Research Problems, American Mathematical Monthly (Mathematical Association of America) 76 (7): 809–810, doi:10.2307/2317879, JSTOR 2317879. 168 CHAPTER 45. HYPOHAMILTONIAN GRAPH

• Lindgren, W. F. (1967), “An inﬁnite class of hypohamiltonian graphs”, American Mathematical Monthly (Mathematical Association of America) 74 (9): 1087–1089, doi:10.2307/2313617, JSTOR 2313617, MR 0224501.

• Máčajová, E.; Škoviera, M. (2007), “Constructing hypohamiltonian snarks with cyclic connectivity 5 and 6”, Electronic Journal of Combinatorics 14 (1): R14.

• Menke, B.; Zamﬁrescu, T. I.; Zamﬁrescu, C. M. (1998), “Intersections of longest cycles in grid graphs”, Journal of Graph Theory 25 (1): 37–52, doi:10.1002/(SICI)1097-0118(199705)25:1<37::AID-JGT2>3.0.CO;2-J.

• Mohanty, S. P.; Rao, D. (1981), “A family of hypo-hamiltonian generalized prisms”, Combinatorics and Graph Theory, Lecture Notes in Mathematics 885, Springer-Verlag, pp. 331–338, doi:10.1007/BFb0092278.

• Park, J.-H.; Lim, H.-S.; Kim, H.-C. (2007), “Panconnectivity and pancyclicity of hypercube-like interconnec- tion networks with faulty elements”, Theoretical Computer Science 377 (1–3): 170–180, doi:10.1016/j.tcs.2007.02.029.

• Robertson, G. N. (1969), Graphs minimal under girth, valency and connectivity constraints, Ph. D. thesis, Waterloo, Ontario: University of Waterloo. • Schauerte, Boris; Zamﬁrescu, C. T. (2006), “Regular graphs in which every pair of points is missed by some longest cycle”, An. Univ. Craiova Ser. Mat. Inform. 33: 154–173, MR 2359901. • Skupień, Z. (1989), “Exponentially many hypohamiltonian graphs”, Graphs, Hypergraphs and Matroids III (Proc. Conf. Kalsk 1988), Zielona Góra: Higher College of Engineering, pp. 123–132. As cited by Skupień (2007).

• Skupień, Z. (2007), “Exponentially many hypohamiltonian snarks”, 6th Czech-Slovak International Symposium on Combinatorics, Graph Theory, Algorithms and Applications, Electronic Notes in Discrete Mathematics 28, pp. 417–424, doi:10.1016/j.endm.2007.01.059. • Sousselier, R. (1963), Berge, C., ed., “Problème no. 29: Le cercle des irascibles”, Problèmes plaisants et délectables, Rev. Franç. Rech. Opérationnelle 7: 405–406. • Steffen, E. (1998), “Classification and characterizations of snarks”, Discrete Mathematics 188 (1–3): 183–203, doi:10.1016/S0012-365X(97)00255-0, MR 1630478. • Steffen, E. (2001), “On bicritical snarks”, Math. Slovaca 51 (2): 141–150, MR 1841443.

• Thomassen, Carsten (1974a), “Hypohamiltonian and hypotraceable graphs”, Discrete Mathematics 9: 91–96, doi:10.1016/0012-365X(74)90074-0, MR 0347682.

• Thomassen, Carsten (1974b), “On hypohamiltonian graphs”, Discrete Mathematics 10 (2): 383–390, doi:10.1016/0012- 365X(74)90128-9, MR 0357226.

• Thomassen, Carsten (1976), “Planar and inﬁnite hypohamiltonian and hypotraceable graphs”, Discrete Math- ematics 14 (4): 377–389, doi:10.1016/0012-365X(76)90071-6, MR 0422086.

• Thomassen, Carsten (1978), “Hypohamiltonian graphs and digraphs”, Theory and applications of graphs (Proc. Internat. Conf., Western Mich. Univ., Kalamazoo, Mich., 1976), Lecture Notes in Mathematics 642, Berlin: Springer-Verlag, pp. 557–571, MR 0499523.

• Thomassen, Carsten (1981), “Planar cubic hypo-Hamiltonian and hypotraceable graphs”, Journal of Combi- natorial Theory, Series B 30: 36–44, doi:10.1016/0095-8956(81)90089-7, MR 0609592.

• Wiener, Gábor; Araya, Makoto (2009), The ultimate question, arXiv:0904.3012. • Zamﬁrescu, C. T.; Zamﬁrescu, T. I. (2007), “A planar hypohamiltonian graph with 48 vertices”, Journal of Graph Theory 55 (4): 338–342, doi:10.1002/jgt.20241.

45.9 External links

• Weisstein, Eric W., “Hypohamiltonian Graph”, MathWorld. Chapter 46

Implication graph

~x2 x0

x6 ~x4 x3

~x5 ~x1 x1 x5

~x3 x4 ~x6

~x0 x2

An implication graph representing the 2-satisﬁability instance (x0∨x2)∧(x0∨¬x3)∧(x1∨¬x3)∧(x1∨¬x4)∧(x2∨¬x4)∧ (x0∨¬x5)∧(x1∨¬x5)∧(x2∨¬x5)∧(x3∨x6)∧(x4∨x6)∧(x5∨x6).

In mathematical logic, an implication graph is a skew-symmetric directed graph G(V, E) composed of vertex set V and directed edge set E. Each vertex in V represents the truth status of a Boolean literal, and each directed edge from vertex u to vertex v represents the material implication “If the literal u is true then the literal v is also true”. Implication graphs were originally used for analyzing complex Boolean expressions.

169 170 CHAPTER 46. IMPLICATION GRAPH

46.1 Applications

A 2-satisfiability instance in conjunctive normal form can be transformed into an implication graph by replacing each of its disjunctions by a pair of implications. An instance is satisfiable if and only if no literal and its negation belong to the same strongly connected component of its implication graph; this characterization can be used to solve 2-satisfiability instances in linear time.[1] In CDCL SAT-solvers, unit propagation can be naturally associated with an implication graph that captures all possible ways of deriving all implied literals from decision literals,[2] which is then used for clause learning.

46.2 References

[1] Aspvall, Bengt; Plass, Michael F.; Tarjan, Robert E. (1979). “A linear-time algorithm for testing the truth of certain quantiﬁed boolean formulas”. Information Processing Letters 8 (3): 121–123. doi:10.1016/0020-0190(79)90002-4.

[2] Paul Beame, Henry Kautz, Ashish Sabharwal (2003). Understanding the Power of Clause Learning (PDF). IJCAI. pp. 1194–1201. Chapter 47

Integral graph

In the mathematical ﬁeld of graph theory, an integral graph is a graph whose spectrum consists entirely of integers. In other words, a graphs is an integral graph if all the eigenvalues of its characteristic polynomial are integers.[1] The notion was introduced in 1974 by Harary and Schwenk.[2]

47.1 Examples

• The complete graph Kn is integral for all n.

• The edgeless graph K¯n is integral for all n. • Among the cubic symmetric graphs the utility graph, the Petersen graph, the Nauru graph and the Desargues graph are integral.

• The Higman–Sims graph, the Hall–Janko graph, the Clebsch graph, the Hoﬀman–Singleton graph, the Shrikhande graph and the Hoﬀman graph are integral.

47.2 References

[1] Weisstein, Eric W., “Integral Graph”, MathWorld.

[2] Harary, F. and Schwenk, A. J. “Which Graphs have Integral Spectra?" In Graphs and Combinatorics (Ed. R. Bari and F. Harary). Berlin: Springer-Verlag, pp. 45–51, 1974.

171 Chapter 48

k-edge-connected graph

In graph theory, a connected graph is k-edge-connected if it remains connected whenever fewer than k edges are removed. The edge-connectivity of a graph is the largest k for which the graph is k-edge-connected.

48.1 Formal deﬁnition

Let G = (V,E) be an arbitrary graph. If subgraph G′ = (V,E \ X) is connected for all X ⊆ E where |X| < k , then G is k-edge-connected. The edge connectivity of G is the maximum value k such that G is k-edge-connected. The smallest set X whose removal disconnects G is a minimum cut in G. The edge connectivity version of Menger’s theorem provides an alternative and equivalent characterization, in terms of edge-disjoint paths in the graph. If every two vertices of G form the endpoints of k paths, no two of which share an edge with each other, then G is k-edge-connected. In one direction this is easy: if a system of paths like this exists, then every set X of fewer than k edges is disjoint from at least one of the paths, and the pair of vertices remains connected to each other even after X is deleted. In the other direction, the existence of a system of paths for each pair of vertices in a graph that cannot be disconnected by the removal of few edges can be proven using the max-ﬂow min-cut theorem from the theory of network ﬂows.

48.2 Related concepts

Minimum vertex degree gives a trivial upper bound on edge-connectivity. That is, if a graph G = (V,E) is k-edge- connected then it is necessary that k ≤ δ(G), where δ(G) is the minimum degree of any vertex v ∈ V. Obviously, deleting all edges incident to a vertex, v, would then disconnect v from the graph. Edge connectivity is the dual concept to girth, the length of the shortest cycle in a graph, in the sense that the girth of a planar graph is the edge connectivity of its dual graph, and vice versa. These concepts are uniﬁed in matroid theory by the girth of a matroid, the size of the smallest dependent set in the matroid. For a graphic matroid, the matroid girth equals the girth of the underlying graph, while for a co-graphic matroid it equals the edge connectivity.[1] The 2-edge-connected graphs can also be characterized by the absence of bridges, by the existence of an ear decomposition, or by Robbins’ theorem according to which these are exactly the graphs that have a strong orientation.[2]

48.3 Computational aspects

There is a polynomial-time algorithm to determine the largest k for which a graph G is k-edge-connected. A simple algorithm would, for every pair (u,v), determine the maximum flow from u to v with the capacity of all edges in G set to 1 for both directions. A graph is k-edge-connected if and only if the maximum flow from u to v is at least k for any pair (u,v), so k is the least u-v-flow among all (u,v).

172 48.4. SEE ALSO 173

If n is the number of vertices in the graph, this simple algorithm would perform O(n2) iterations of the Maximum flow problem, which can be solved in O(n3) time. Hence the complexity of the simple algorithm described above is O(n5) in total. An improved algorithm will solve the maximum flow problem for every pair (u,v) where u is arbitrarily fixed while v varies over all vertices. This reduces the complexity to O(n4) and is sound since, if a cut of capacity less than k exists, it is bound to separate u from some other vertex. It can be further improved by an algorithm of Gabow that runs in worst case O(n3) time. [3] The Karger–Stein variant of Karger’s algorithm provides a faster randomized algorithm for determining the connectivity, with expected runtime O(n2 log3 n) .[4] A related problem: finding the minimum k-edge-connected subgraph of G (that is: select as few as possible edges in G that your selection is k-edge-connected) is NP-hard for k ≥ 2 .[5]

48.4 See also

• k-vertex-connected graph • Connectivity (graph theory)

• Matching preclusion

48.5 References

[1] Cho, Jung Jin; Chen, Yong; Ding, Yu (2007), “On the (co)girth of a connected matroid”, Discrete Applied Mathematics 155 (18): 2456–2470, doi:10.1016/j.dam.2007.06.015, MR 2365057.

[2] Robbins, H. E. (1939). “A theorem on graphs, with an application to a problem on traﬃc control”. American Mathematical Monthly 46: 281–283. doi:10.2307/2303897. JSTOR 2303897.

[3] Harold N. Gabow. A matroid approach to ﬁnding edge connectivity and packing arborescences. J. Comput. Syst. Sci., 50(2):259–273, 1995.

[4] Karger, David R.; Stein, Cliﬀord (1996). “A new approach to the minimum cut problem” (PDF). Journal of the ACM 43 (4): 601. doi:10.1145/234533.234534.

[5] M.R. Garey and D.S. Johnson. Computers and Intractability: a Guide to the Theory of NP-Completeness. Freeman, San Francisco, CA, 1979. Chapter 49

K-tree

The Goldner–Harary graph, an example of a planar 3-tree.

In graph theory, a k-tree is a chordal graph all of whose maximal cliques are the same size k + 1 and all of whose minimal clique separators are also all the same size k.[1] The k-trees are exactly the maximal graphs with a given treewidth, graphs to which no more edges can be added without increasing their treewidth. The graphs that have treewidth at most k are exactly the subgraphs of k-trees, and for this reason they are called partial k-trees.[2] Every k-tree may be formed by starting with a (k + 1)-vertex complete graph and then repeatedly adding vertices in such a way that each added vertex has exactly k neighbors that form a clique.[1][2] Certain k-trees with k ≥ 3 are also the graphs formed by the edges and vertices of stacked polytopes, polytopes formed by starting from a simplex and then repeatedly gluing simplices onto the faces of the polytope; this gluing process mimics the construction of k-trees by adding vertices to a clique.[3] Every stacked polytope forms a k-tree in this way, but not every k-tree comes from a stacked polytope: a k-tree is the graph of a stacked polytope if and only if no three (k + 1)-vertex cliques have k vertices in common.[4] 1-trees are the same as unrooted trees. 2-trees are maximal series-parallel graphs,[5] and include also the maximal outerplanar graphs. Planar 3-trees are also known as Apollonian networks.[6] In higher-dimensional geometry, the stacked polytopes have graphs that are k-trees.[7]

174 49.1. REFERENCES 175

49.1 References

[1] Patil, H. P. (1986), “On the structure of k-trees”, Journal of Combinatorics, Information and System Sciences 11 (2-4): 57–64, MR 966069.

[2] Nešetřil, Jaroslav; Ossona de Mendez, Patrice (2008), “Structural Properties of Sparse Graphs” (PDF), in Grötschel, Mar- tin; Katona, Gyula O. H., Building Bridges: between Mathematics and Computer Science, Bolyai Society Mathematical Studies 19, Springer-Verlag, p. 390, ISBN 978-3-540-85218-6.

[3] Below, Alexander; De Loera, Jesús A.; Richter-Gebert, Jürgen. “The Complexity of Finding Small Triangulations of Convex 3-Polytopes”. arXiv:math/0012177..

[4] Kleinschmidt, Peter (1 December 1976). “Eine graphentheoretische Kennzeichnung der Stapelpolytope”. Archiv der Math- ematik 27 (1): 663–667. doi:10.1007/BF01224736.

[5] Hwang, Frank; Richards, Dana; Winter, Pawel (1992), The Steiner Tree Problem, Annals of Discrete Mathematics (North- Holland Mathematics Studies) 53, Elsevier, p. 177, ISBN 978-0-444-89098-6.

[6] Distances in random Apollonian network structures, talk slides by Olivier Bodini, Alexis Darrasse, and Michèle Soria from a talk at FPSAC 2008, accessed 2011-03-06.

[7] Koch, Etan; Perles, Micha A. (1976), “Covering eﬃciency of trees and k-trees”, Proceedings of the Seventh Southeastern Conference on Combinatorics, Graph Theory, and Computing (Louisiana State Univ., Baton Rouge, La., 1976), Utilitas Math., Winnipeg, Man., pp. 391–420. Congressus Numerantium, No. XVII, MR 0457265. See in particular p. 420. Chapter 50

k-vertex-connected graph

In graph theory, a connected graph G is said to be k-vertex-connected (or k-connected) if it has more than k vertices and remains connected whenever fewer than k vertices are removed. The vertex-connectivity, or just connectivity, of a graph is the largest k for which the graph is k-vertex-connected.

50.1 Deﬁnitions

A graph (other than a complete graph) has connectivity k if k is the size of the smallest subset of vertices such that the graph becomes disconnected if you delete them.[1] Complete graphs are not included in this version of the definition since they cannot be disconnected by deleting vertices. The complete graph with n vertices has connectivity n − 1, as implied by the first definition. An equivalent definition is that a graph with at least two vertices is k-connected if, for every pair of its vertices, it is possible to find k vertex-independent paths connecting these vertices; see Menger’s theorem (Diestel 2005, p. 55). This definition produces the same answer, n − 1, for the connectivity of the complete graph Kn.[1] A 1-connected graph is called connected; a 2-connected graph is called biconnected. A 3-connected graph is called triconnected.

50.2 Applications

50.2.1 Polyhedral Combinatorics

The 1-skeleton of any k-dimensional convex polytope forms a k-vertex-connected graph (Balinski’s theorem, Balinski 1961). As a partial converse, Steinitz’s theorem states that any 3-vertex-connected planar graph forms the skeleton of a convex polyhedron.

50.3 Computational complexity

The vertex-connectivity of an input graph G can be computed in polynomial time in the following way[2] consider all possible pairs (s, t) of nonadjacent nodes to disconnect, using Menger’s theorem to justify that the minimal-size separator for (s, t) is the number of pairwise vertex-independent paths between them, encode the input by doubling each vertex as an edge to reduce to a computation of the number of pairwise edge-independent paths, and compute the maximum number of such paths by computing the maximum flow in the graph between s and t with capacity 1 to each edge, noting that a flow of k in this graph corresponds, by the integral flow theorem, to k pairwise edge-independent paths from s to t .

176 50.4. SEE ALSO 177

50.4 See also

• k-edge-connected graph

• Connectivity (graph theory) • Menger’s theorem

• Structural cohesion

• Tutte embedding • Vertex separator

50.5 Notes

[1] Schrijver, Combinatorial Optimization, Springer

[2] The algorithm design manual, p 506, and Computational discrete mathematics: combinatorics and graph theory with Math- ematica, p. 290-291

50.6 References

• Balinski, M. L. (1961), “On the graph structure of convex polyhedra in n-space”, Paciﬁc Journal of Mathematics 11 (2): 431–434, doi:10.2140/pjm.1961.11.431.

• Diestel, Reinhard (2005), Graph Theory (3rd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540- 26183-4. Chapter 51

Laman graph

The Moser spindle, a planar Laman graph drawn as a pointed pseudotriangulation In graph theory, the Laman graphs are a family of sparse graphs describing the minimally rigid systems of rods and joints in the plane. Formally, a Laman graph is a graph on n vertices such that, for all k, every k-vertex subgraph has at most 2k − 3 edges, and such that the whole graph has exactly 2n − 3 edges. Laman graphs are named after Gerard Laman, of the University of Amsterdam, who in 1970 used them to characterize rigid planar structures.[1]

51.1 Rigidity

Laman graphs arise in rigidity theory: if one places the vertices of a Laman graph in the Euclidean plane, in general position, there will in general be no simultaneous motion of all the points, other than Euclidean congruences, that preserves the lengths of all the graph edges. A graph is rigid in this sense if and only if it has a Laman subgraph that spans all of its vertices. Thus, the Laman graphs are exactly the minimally rigid graphs, and they form the bases of the two-dimensional rigidity matroids. If n points in the plane are given, then there are 2n degrees of freedom in their placement (each point has two independent coordinates), but a rigid graph has only three degrees of freedom (the position of a single one of its vertices and the rotation of the remaining graph around that vertex). Intuitively, adding an edge of ﬁxed length to a graph reduces its number of degrees of freedom by one, so the 2n − 3 edges in a Laman graph reduce the 2n degrees of freedom of the initial point placement to the three degrees of freedom of a rigid graph. However, not every graph with 2n − 3 edges is rigid; the condition in the deﬁnition of a Laman graph that no subgraph can have too many edges

178 51.2. PLANARITY 179

The complete bipartite graph K3,3, a non-planar Laman graph

ensures that each edge contributes to reducing the overall number of degrees of freedom, and is not wasted within a subgraph that is already itself rigid due to its other edges.

51.2 Planarity

A pointed pseudotriangulation is a planar straight-line drawing of a graph, with the properties that the outer face is convex, that every bounded face is a pseudotriangle, a polygon with only three convex vertices, and that the edges incident to every vertex span an angle of less than 180 degrees. The graphs that can be drawn as pointed pseudotriangulations are exactly the planar Laman graphs.[2] However, Laman graphs have planar embeddings that are not pseudotriangulations, and there are Laman graphs that are not planar, such as the utility graph K₃,₃.

51.3 Sparsity

Lee & Streinu (2008) and Streinu & Theran (2009) deﬁne a graph as being (k, l) -sparse if every nonempty subgraph with n vertices has at most kn − l edges, and (k, l) -tight if it is (k, l) -sparse and has exactly kn − l edges. Thus, in their notation, the Laman graphs are exactly the (2,3)-tight graphs, and the subgraphs of the Laman graphs are exactly the (2,3)-sparse graphs. The same notation can be used to describe other important families of sparse graphs, including trees, pseudoforests, and graphs of bounded arboricity.[3][4] Based on this characterization, it is possible to recognize n-vertex Laman graphs in time O(n2), by simulating a “pebble game” that begins with a graph with n vertices and no edges, with two pebbles placed on each vertex, and 180 CHAPTER 51. LAMAN GRAPH

performs a sequence of the following two kinds of steps to create all of the edges of the graph:

• Create a new directed edge connecting any two vertices that both have two pebbles, and remove one pebble from the start vertex of the new edge. • If an edge points from a vertex u with at most one pebble to another vertex v with at least one pebble, move a pebble from v to u and reverse the edge.

If these operations can be used to construct an orientation of the given graph,√ then it is necessarily (2,3)-sparse, and vice versa. However, faster algorithms are possible, running in time O(n3/2 log n) , based on testing whether doubling one edge of the given graph results in a multigraph that is (2,2)-tight (equivalently, whether it can be decomposed into two edge-disjoint spanning trees) and then using this decomposition to check whether the given graph is a Laman graph.[5]

51.4 Henneberg construction

Henneberg construction of the Moser spindle

Long before Laman’s work, Lebrecht Henneberg characterized the two-dimensional minimally rigid graphs (that is, the Laman graphs) in a diﬀerent way.[6] Henneberg showed that the minimally rigid graphs on two or more vertices are exactly the graphs that can be obtained, starting from a single edge, by a sequence of operations of the following two types:

1. Add a new vertex to the graph, together with edges connecting it to two previously existing vertices. 2. Subdivide an edge of the graph, and add an edge connecting the newly formed vertex to a third previously existing vertex.

A sequence of these operations that forms a given graph is known as a Henneberg construction of the graph. For instance, the complete bipartite graph K₃,₃ may be formed using the ﬁrst operation to form a triangle and then applying the second operation to subdivide each edge of the triangle and connect each subdivision point with the opposite triangle vertex. 51.5. REFERENCES 181

51.5 References

[1] Laman, G. (1970), “On graphs and the rigidity of plane skeletal structures”, J. Engineering Mathematics 4 (4): 331–340, doi:10.1007/BF01534980, MR 0269535.

[2] Haas, Ruth; Orden, David; Rote, Günter; Santos, Francisco; Servatius, Brigitte; Servatius, Herman; Souvaine, Diane; Streinu, Ileana; Whiteley, Walter (2005), “Planar minimally rigid graphs and pseudo-triangulations”, Computational Ge- ometry Theory and Applications 31 (1–2): 31–61, doi:10.1016/j.comgeo.2004.07.003, MR 2131802.

[3] Lee, Audrey; Streinu, Ileana (2008), “Pebble game algorithms and sparse graphs”, Discrete Mathematics 308 (8): 1425– 1437, doi:10.1016/j.disc.2007.07.104, MR 2392060.

[4] Streinu, I.; Theran, L. (2009), “Sparse hypergraphs and pebble game algorithms”, European Journal of Combinatorics 30 (8): 1944–1964, arXiv:math/0703921, doi:10.1016/j.ejc.2008.12.018.

[5] Daescu, O.; Kurdia, A. (2009), “Towards an optimal algorithm for recognizing Laman graphs”, Proc. 42nd Hawaii Inter- national Conference on System Sciences (HICSS '09), IEEE, pp. 1–10, doi:10.1109/HICSS.2009.470.

[6] Henneberg, L. (1911), Die graphische Statik der starren Systeme, Leipzig Chapter 52

Lattice graph

A lattice graph, mesh graph, or grid graph, is a graph whose drawing, embedded in some Euclidean space Rn, forms a regular tiling. This implies that the group of bijective transformations that send the graph to itself is a lattice in the group-theoretical sense. Typically, no clear distinction is made between such a graph in the more abstract sense of graph theory, and its drawing in space (often the plane or 3D space). This type of graph may more shortly be called just a lattice, mesh, or grid. Moreover, these terms are also commonly used for a ﬁnite section of the inﬁnite graph, as in “an 8×8 square grid”. The term lattice graph has also been given in the literature to various other kinds of graphs with some regular structure, such as the Cartesian product of a number of complete graphs.[1]

52.1 Square grid graph

A common type of a lattice graph (known under diﬀerent names, such as square grid graph) is the graph whose vertices correspond to the points in the plane with integer coordinates, x-coordinates being in the range 1,..., n, y- coordinates being in the range 1,..., m, and two vertices are connected by an edge whenever the corresponding points are at distance 1. In other words, it is a unit distance graph for the described point set.[2]

52.1.1 Properties

A square grid graph is a Cartesian product of graphs, namely, of two path graphs with n - 1 and m - 1 edges.[2] Since a path graph is a median graph, the latter fact implies that the square grid graph is also a median graph. All grid graphs are bipartite, which is easily veriﬁed by the fact that one can color the vertices in a checkerboard fashion. A path graph may also be considered to be a grid graph on the grid n times 1. A 2x2 grid graph is a 4-cycle.[2] Every planar graph H is a minor of the h×h-grid, where h = 2|V(H)| + 4|E(H)|[3].

52.2 Other kinds

A triangular grid graph is a graph that corresponds to a triangular grid. A Hanan grid graph for a ﬁnite set of points in the plane is produced by the grid obtained by intersections of all vertical and horizontal lines through each point of the set. The rook’s graph (the graph that represents all legal moves of the rook chess piece on a chessboard) is also sometimes called the lattice graph.

182 52.3. SEE ALSO 183

52.3 See also

• Lattice path

• Pick’s theorem • Integer triangles in a 2D lattice

52.4 References

[1] Weisstein, Eric W., “Lattice graph”, MathWorld.

[2] Weisstein, Eric W., “Grid graph”, MathWorld.

[3] Robertson, N.; Seymour, P.; Thomas, R. (November 1994). “Quickly Excluding a Planar Graph”. Journal of Combinatorial Theory, Series B 62 (2): 323–348. doi:10.1006/jctb.1994.1073. Chapter 53

Leaf power

In the mathematical area of graph theory, a tree T is a k-leaf root of a graph G = (V,E) if V is contained in the set of leaves of T and for vertices x,y in V, xy is an edge in E if and only if their distance in T is at most k. Then G is called a k-leaf power. A graph is a leaf power if it is a k-leaf power for some k.[1] This is an important class of graphs based on phylogeny. Since powers of strongly chordal graphs are strongly chordal and trees are strongly chordal, it follows that leaf powers are strongly chordal graphs.[2] Actually, leaf powers form a proper subclass of strongly chordal graphs; a graph is a leaf power if and only if it is a ﬁxed tolerance NeST graph[3]and such graphs are a proper subclass of strongly chordal graphs. [4]

It is easy to see that a graph is a 2-leaf power if and only if it is the disjoint union of cliques (i.e., a P3-free graph). A graph is a 3-leaf power if and only if it is a (bull,dart,gem)-free chordal graph.[5] Based on this characterization and similar ones, 3-leaf powers can be recognized in linear time.[6] Characterizations of 4-leaf powers are given in [7] which also enable linear time recognition. For k > 5 the recognition problem of k-leaf powers is open, and likewise it is an open problem whether leaf powers can be recognized in polynomial time. In[8] it is shown that interval graphs and the larger class of rooted directed path graphs are leaf powers.

53.1 Notes

[1] Nishimura, Ragde & Thilikos (2002)

[2] Dahlhaus & Duchet (1987),Lubiw (1987),Raychaudhuri (1992)

[3] Brandstädt et al. (2010),Hayward, Kearney & Malton (2002)

[4] Broin & Lowe (1986),Bibelnieks & Dearing (2006)

[5] Dom et al. (2006),Rautenbach (2006)

[6] Brandstädt & Le (2006)

[7] Rautenbach (2006),Brandstädt, Le & Sritharan (2008)

[8] Brandstädt et al. (2010)

53.2 References

• Bibelnieks, E.; Dearing, P.M. (2006), “Neighborhood subtree tolerance graphs”, Discrete Applied Mathematics 98: 133–138.

184 53.2. REFERENCES 185

• Brandstädt, Andreas; Hundt, Christian; Mancini, Federico; Wagner, Peter (2010), “Rooted directed path graphs are leaf powers”, Discrete Mathematics 310: 897–910. • Brandstädt, Andreas; Le, Van Bang (2006), “Structure and linear time recognition of 3-leaf powers”, Information Processing Letters 98: 133–138. • Brandstädt, Andreas; Le, Van Bang; Sritharan, R. (2008), “Structure and linear time recognition of 4-leaf powers”, ACM Transactions on Algorithms 5: Article 11. • Brandstädt, Andreas; Le, Van Bang; Spinrad, Jeremy (1999), Graph Classes: A Survey, SIAM Monographs on Discrete Mathematics and Applications, ISBN 0-89871-432-X. • Broin, M.W.; Lowe, T.J. (1986), “Neighborhood subtree tolerance graphs”, SIAM J. Algebraic and Discrete Meth. 7: 348–357. • Dahlhaus, E.; Duchet, P. (1987), “On strongly chordal graphs”, Ars Combinatoria 24 B: 23–30.

• Dahlhaus, E.; Manuel, P. D.; Miller, M. (1998), “A characterization of strongly chordal graphs”, Discrete Mathematics 187 (1–3): 269–271, doi:10.1016/S0012-365X(97)00268-9. • Dom, M.; Guo, J.; Hüﬀner, F.; Niedermeier, R. (2006), “Error compensation in leaf root problems”, Algorithmica 44: 363–381. • Farber, M. (1983), “Characterizations of strongly chordal graphs”, Discrete Mathematics 43 (2–3): 173–189, doi:10.1016/0012-365X(83)90154-1. • Hayward, R.B.; Kearney, P.E.; Malton, A. (2002), “NeST graphs”, Discrete Applied Mathematics 121: 139– 153. • Lubiw, A. (1987), “Doubly lexical orderings of matrices”, SIAM Journal on Computing 16 (5): 854–879, doi:10.1137/0216057. • McKee, T. A. (1999), “A new characterization of strongly chordal graphs”, Discrete Mathematics 205 (1–3): 245–247, doi:10.1016/S0012-365X(99)00107-7. • Nishimura, N.; Ragde, P.; Thilikos, D.M. (2002), “On graph powers for leaf-labeled trees”, Journal of Algo- rithms 42: 69–108. • Rautenbach, D. (2006), “Some remarks about leaf roots”, Discrete Mathematics 306: 1456–1461.

• Raychaudhuri, A. (1992), “On powers of strongly chordal graphs and circular arc graphs”, Ars Combinatoria 34: 147–160. Chapter 54

Line graph

This article is about the mathematical concept. For the statistical presentation method, see line chart. Not to be confused with linear graph.

In the mathematical discipline of graph theory, the line graph of an undirected graph G is another graph L(G) that represents the adjacencies between edges of G. The name line graph comes from a paper by Harary & Norman (1960) although both Whitney (1932) and Krausz (1943) used the construction before this.[1] Other terms used for the line graph include the covering graph, the derivative, the edge-to-vertex dual, the conjugate, the representative graph, and the ϑ-obrazom,[1] as well as the edge graph, the interchange graph, the adjoint graph, and the derived graph.[2] Hassler Whitney (1932) proved that with one exceptional case the structure of a connected graph G can be recovered completely from its line graph.[3] Many other properties of line graphs follow by translating the properties of the underlying graph from vertices into edges, and by Whitney’s theorem the same translation can also be done in the other direction. Line graphs are claw-free, and the line graphs of bipartite graphs are perfect. Line graphs can be characterized by nine forbidden subgraphs, and can be recognized in linear time. Various generalizations of line graphs have also been studied, including the line graphs of line graphs, line graphs of multigraphs, line graphs of hypergraphs, and line graphs of weighted graphs.

54.1 Formal deﬁnition

Given a graph G, its line graph L(G) is a graph such that

• each vertex of L(G) represents an edge of G; and

• two vertices of L(G) are adjacent if and only if their corresponding edges share a common endpoint (“are incident”) in G.

That is, it is the intersection graph of the edges of G, representing each edge by the set of its two endpoints.[2]

54.2 Example

The following ﬁgures show a graph (left, with blue vertices) and its line graph (right, with green vertices). Each vertex of the line graph is shown labeled with the pair of endpoints of the corresponding edge in the original graph. For instance, the green vertex on the right labeled 1,3 corresponds to the edge on the left between the blue vertices 1 and 3. Green vertex 1,3 is adjacent to three other green vertices: 1,4 and 1,2 (corresponding to edges sharing the endpoint 1 in the blue graph) and 4,3 (corresponding to an edge sharing the endpoint 3 in the blue graph).

• Graph G

186 54.3. PROPERTIES 187

• Vertices in L(G) constructed from edges in G

• Added edges in L(G)

• The line graph L(G)

54.3 Properties

54.3.1 Translated properties of the underlying graph

Properties of a graph G that depend only on adjacency between edges may be translated into equivalent properties in L(G) that depend on adjacency between vertices. For instance, a matching in G is a set of edges no two of which are adjacent, and corresponds to a set of vertices in L(G) no two of which are adjacent, that is, an independent set.[4] Thus,

• The line graph of a connected graph is connected. If G is connected, it contains a path connecting any two of its edges, which translates into a path in L(G) containing any two of the vertices of L(G). However, a graph G that has some isolated vertices, and is therefore disconnected, may nevertheless have a connected line graph.[5]

• A line graph has an articulation point if and only if the underlying graph has a bridge for which neither endpoint has degree one.[2]

• For a graph G with n vertices and m edges, the number of vertices of the line graph L(G) is m, and the number of edges of L(G) is half the sum of the squares of the degrees of the vertices in G, minus m.[6]

• A maximum independent set in a line graph corresponds to maximum matching in the original graph. Since maximum matchings may be found in polynomial time, so may the maximum independent sets of line graphs, despite the hardness of the maximum independent set problem for more general families of graphs.[4]

• The edge chromatic number of a graph G is equal to the vertex chromatic number of its line graph L(G).[7]

• The line graph of an edge-transitive graph is vertex-transitive. This property can be used to generate families of graphs that (like the Petersen graph) are vertex-transitive but are not Cayley graphs: if G is an edge-transitive graph that has at least ﬁve vertices, is not bipartite, and has odd vertex degrees, then L(G) is a vertex-transitive non-Cayley graph.[8]

• If a graph G has an Euler cycle, that is, if G is connected and has an even number of edges at each vertex, then the line graph of G is Hamiltonian. However, not all Hamiltonian cycles in line graphs come from Euler cycles in this way; for instance, the line graph of a Hamiltonian graph G is itself Hamiltonian, regardless of whether G is also Eulerian.[9]

• If two simple graphs are isomorphic then their line graphs are also isomorphic. The Whitney graph isomorphism theorem provides a converse to this for every but one graph.

• In the context of complex network theory, the line graph of a random network preserves many of the properties of the network such as the small-world property (the existence of short paths between all pairs of vertices) and the shape of its degree distribution.[10] Evans & Lambiotte (2009) observe that any method for ﬁnding vertex clusters in a complex network can be applied to the line graph and used to cluster its edges instead.

54.3.2 Whitney isomorphism theorem

If the line graphs of two connected graphs are isomorphic, then the underlying graphs are isomorphic, except in the case of the triangle graph K3 and the claw K₁,₃, which have isomorphic line graphs but are not themselves isomorphic.[3]

As well as K3 and K₁,₃, there are some other exceptional small graphs with the property that their line graph has a higher degree of symmetry than the graph itself. For instance, the diamond graph K₁,₁,₂ (two triangles sharing an edge) has four graph automorphisms but its line graph K₁,₂,₂ has eight. In the illustration of the diamond graph shown, rotating the graph by 90 degrees is not a symmetry of the graph, but is a symmetry of its line graph. However, all 188 CHAPTER 54. LINE GRAPH

The diamond graph, an exception to the strong Whitney theorem such exceptional cases have at most four vertices. A strengthened version of the Whitney isomorphism theorem states that, for connected graphs with more than four vertices, there is a one-to-one correspondence between isomorphisms of the graphs and isomorphisms of their line graphs.[11] Analogues of the Whitney isomorphism theorem have been proven for the line graphs of multigraphs, but are more complicated in this case.[12]

54.3.3 Strongly regular and perfect line graphs

The line graph of the complete graph Kn is also known as the triangular graph, the Johnson graph J(n,2), or the complement of the Kneser graph KGn,₂. Triangular graphs are characterized by their spectra, except for n = 8.[13] They may also be characterized (again with the exception of K8) as the strongly regular graphs with parameters srg(n(n − 1)/2, 2(n − 2), n − 2, 4).[14] The three strongly regular graphs with the same parameters and spectrum as L(K8) are the Chang graphs, which may be obtained by graph switching from L(K8). The line graph of a bipartite graph is perfect (see König’s theorem). The line graphs of bipartite graphs form one of the key building blocks of perfect graphs, used in the proof of the strong perfect graph theorem.[15] A special case of these graphs are the rook’s graphs, line graphs of complete bipartite graphs. Like the line graphs of complete graphs, they can be characterized with one exception by their numbers of vertices, numbers of edges, and number of shared 54.3. PROPERTIES 189

A line perfect graph. The edges in each biconnected component are colored black if the component is bipartite, blue if the component is a tetrahedron, and red if the component is a book of triangles.

neighbors for adjacent and non-adjacent points. The one exceptional case is L(K₄,₄), which shares its parameters with the Shrikhande graph. When both sides of the bipartition have the same number of vertices, these graphs are again strongly regular.[16] More generally, a graph G is said to be line perfect if L(G) is a perfect graph. The line perfect graphs are exactly the graphs that do not contain a simple cycle of odd length greater than three.[17] Equivalently, a graph is line perfect if and only if each of its biconnected components is either bipartite or of the form K4 (the tetrahedron) or K₁,₁,n (a book of one or more triangles all sharing a common edge).[18] Every line perfect graph is itself perfect.[19]

54.3.4 Other related graph families

All line graphs are claw-free graphs, graphs without an induced subgraph in the form of a three-leaf tree.[20] As with claw-free graphs more generally, every connected line graph L(G) with an even number of edges has a perfect matching;[21] equivalently, this means that if the underlying graph G has an even number of edges, its edges can be partitioned into two-edge paths. The line graphs of trees are exactly the claw-free block graphs.[22] These graphs have been used to solve a problem in extremal graph theory, of constructing a graph with a given number of edges and vertices whose largest tree induced as a subgraph is as small as possible.[23] All eigenvalues of the adjacency matrix of a line graph are at least −2.For this reason, the graphs whose eigenvalues have this property have been called generalized line graphs.[24] 190 CHAPTER 54. LINE GRAPH

54.4 Characterization and recognition

54.4.1 Clique partition

Partition of a line graph into cliques

For an arbitrary graph G, and an arbitrary vertex v in G, the set of edges incident to v corresponds to a clique in the line graph L(G). The cliques formed in this way partition the edges of L(G). Each vertex of L(G) belongs to exactly two of them (the two cliques corresponding to the two endpoints of the corresponding edge in G). The existence of such a partition into cliques can be used to characterize the line graphs: A graph L is the line graph of some other graph or multigraph if and only if it is possible to ﬁnd a collection of cliques in L (allowing some of the cliques to be single vertices) that partition the edges of L, such that each vertex of L belongs to exactly two of the cliques.[20] It is the line graph of a graph (rather than a multigraph) if this set of cliques satisﬁes the additional condition that no two vertices of L are both in the same two cliques. Given such a family of cliques, the underlying graph G for which L is the line graph can be recovered by making one vertex in G for each clique, and an edge in G for each vertex in L with its endpoints being the two cliques containing the vertex in L. By the strong version of Whitney’s isomorphism theorem, if the underlying graph G has more than four vertices, there can be only one partition of this type. For example, this characterization can be used to show that the following graph is not a line graph:

In this example, the edges going upward, to the left, and to the right from the central degree-four vertex do not have any cliques in common. Therefore, any partition of the graph’s edges into cliques would have to have at least one clique for each of these three edges, and these three cliques would all intersect in that central vertex, violating the requirement that each vertex appear in exactly two cliques. Thus, the graph shown is not a line graph. 54.4. CHARACTERIZATION AND RECOGNITION 191

54.4.2 Forbidden subgraphs

The nine minimal non-line graphs, from Beineke’s forbidden-subgraph characterization of line graphs. A graph is a line graph if and only if it does not contain one of these nine graphs as an induced subgraph.

An alternative characterization of line graphs was proven by Beineke (1970) (and reported earlier without proof by Beineke (1968)). He showed that there are nine minimal graphs that are not line graphs, such that any graph that is not a line graph has one of these nine graphs as an induced subgraph. That is, a graph is a line graph if and only if no subset of its vertices induces one of these nine graphs. In the example above, the four topmost vertices induce a claw (that is, a complete bipartite graph K₁,₃), shown on the top left of the illustration of forbidden subgraphs. Therefore, by Beineke’s characterization, this example cannot be a line graph. For graphs with minimum degree at least 5, only the six subgraphs in the left and right columns of the ﬁgure are needed in the characterization.[25] Line graphs of multigraphs may be similarly characterized by three of Beineke’s nine forbidden subgraphs.[26]

54.4.3 Algorithms

Roussopoulos (1973) and Lehot (1974) described linear time algorithms for recognizing line graphs and reconstruct- ing their original graphs. Sysło (1982) generalized these methods to directed graphs. Degiorgi & Simon (1995) described an eﬃcient data structure for maintaining a dynamic graph, subject to vertex insertions and deletions, and maintaining a representation of the input as a line graph (when it exists) in time proportional to the number of changed edges at each step. The algorithms of Roussopoulos (1973) and Lehot (1974) are based on characterizations of line graphs involving odd triangles (triangles in the line graph with the property that there exists another vertex adjacent to an odd number of triangle vertices). However, the algorithm of Degiorgi & Simon (1995) uses only Whitney’s isomorphism theorem. It is complicated by the need to recognize deletions that cause the remaining graph to become a line graph, but when specialized to the static recognition problem only insertions need to be performed, and the algorithm performs the following steps: 192 CHAPTER 54. LINE GRAPH

• Construct the input graph L by adding vertices one at a time, at each step choosing a vertex to add that is adjacent to at least one previously-added vertex. While adding vertices to L, maintain a graph G for which L = L(G); if the algorithm ever fails to ﬁnd an appropriate graph G, then the input is not a line graph and the algorithm terminates. • When adding a vertex v to a graph L(G) for which G has four or fewer vertices, it might be the case that the line graph representation is not unique. But in this case, the augmented graph is small enough that a representation of it as a line graph can be found by a brute force search in constant time. • When adding a vertex v to a larger graph L that equals the line graph of another graph G, let S be the subgraph of G formed by the edges that correspond to the neighbors of v in L. Check that S has a vertex cover consisting of one vertex or two non-adjacent vertices. If there are two vertices in the cover, augment G by adding an edge (corresponding to v) that connects these two vertices. If there is only one vertex in the cover, then add a new vertex to G, adjacent to this vertex.

Each step either takes constant time, or involves ﬁnding a vertex cover of constant size within a graph S whose size is proportional to the number of neighbors of v. Thus, the total time for the whole algorithm is proportional to the sum of the numbers of neighbors of all vertices, which (by the handshaking lemma) is proportional to the number of input edges.

54.5 Iterating the line graph operator

van Rooij & Wilf (1965) consider the sequence of graphs

G, L(G),L(L(G)),L(L(L(G))),....

They show that, when G is a ﬁnite connected graph, only four behaviors are possible for this sequence:

• If G is a cycle graph then L(G) and each subsequent graph in this sequence is isomorphic to G itself. These are the only connected graphs for which L(G) is isomorphic to G.[27] • If G is a claw K₁,₃, then L(G) and all subsequent graphs in the sequence are triangles. • If G is a path graph then each subsequent graph in the sequence is a shorter path until eventually the sequence terminates with an empty graph. • In all remaining cases, the sizes of the graphs in this sequence eventually increase without bound.

If G is not connected, this classiﬁcation applies separately to each component of G. For connected graphs that are not paths, all suﬃciently high numbers of iteration of the line graph operation produce graphs that are Hamiltonian.[28]

54.6 Generalizations

54.6.1 Medial graphs and convex polyhedra

Main article: Medial graph

When a planar graph G has maximum vertex degree three, its line graph is planar, and every planar embedding of G can be extended to an embedding of L(G). However, there exist planar graphs with higher degree whose line graphs are nonplanar. These include, for example, the 5-star K₁,₅, the gem graph formed by adding two non-crossing diagonals within a regular pentagon, and all convex polyhedra with a vertex of degree four or more.[29] An alternative construction, the medial graph, coincides with the line graph for planar graphs with maximum degree three, but is always planar. It has the same vertices as the line graph, but potentially fewer edges: two vertices of 54.6. GENERALIZATIONS 193

the medial graph are adjacent if and only if the corresponding two edges are consecutive on some face of the planar embedding. The medial graph of the dual graph of a plane graph is the same as the medial graph of the original plane graph.[30] For regular polyhedra or simple polyhedra, the medial graph operation can be represented geometrically by the operation of cutting oﬀ each vertex of the polyhedron by a plane through the midpoints of all its incident edges.[31] This operation is known variously as the second truncation,[32] degenerate truncation,[33] or rectiﬁcation.[34]

54.6.2 Total graphs

The total graph T(G) of a graph G has as its vertices the elements (vertices or edges) of G, and has an edge between two elements whenever they are either incident or adjacent. The total graph may also be obtained by subdividing each edge of G and then taking the square of the subdivided graph.[35]

54.6.3 Multigraphs

The concept of the line graph of G may naturally be extended to the case where G is a multigraph. In this case, the characterizations of these graphs can be simpliﬁed: the characterization in terms of clique partitions no longer needs to prevent two vertices from belonging to the same to cliques, and the characterization by forbidden graphs has fewer forbidden graphs.[26] However, for multigraphs, there are larger numbers of pairs of non-isomorphic graphs that have the same line graphs. For instance a complete bipartite graph K₁,n has the same line graph as the dipole graph and Shannon multigraph with the same number of edges. Nevertheless, analogues to Whitney’s isomorphism theorem can still be derived in this case.[12]

54.6.4 Line digraphs

001 011

0 1 000 101 010 111

100 110

01 001 01 011

00 0 1 11 00 000 101 010 111 11

10 100 10 110

00 11

Construction of the de Bruijn graphs as iterated line digraphs

It is also possible to generalize line graphs to directed graphs.[36] If G is a directed graph, its directed line graph or line digraph has one vertex for each edge of G. Two vertices representing directed edges from u to v and from w to 194 CHAPTER 54. LINE GRAPH

x in G are connected by an edge from uv to wx in the line digraph when v = w. That is, each edge in the line digraph of G represents a length-two directed path in G. The de Bruijn graphs may be formed by repeating this process of forming directed line graphs, starting from a complete directed graph.[37]

54.6.5 Weighted line graphs

In a line graph L(G), each vertex of degree k in the original graph G creates k(k-1)/2 edges in the line graph. For many types of analysis this means high degree nodes in G are over represented in the line graph L(G). For instance consider a random walk on the vertices of the original graph G. This will pass along some edge e with some frequency f. On the other hand this edge e is mapped to a unique vertex, say v, in the line graph L(G). If we now perform the same type of random walk on the vertices of the line graph, the frequency with which v is visited can be completely different from f. If our edge e in G was connected to nodes of degree O(k), it will be traversed O(k2) more frequently in the line graph L(G). Put another way, the Whitney graph isomorphism theorem guarantees that the line graph almost always encodes the topology of the original graph G faithfully but it does not guarantee that dynamics on these two graphs have a simple relationship. One solution is to construct a weighted line graph, that is, a line graph with weighted edges. There are several natural ways to do this.[38] For instance if edges d and e in the graph G are incident at a vertex v with degree k, then in the line graph L(G) the edge connecting the two vertices d and e can be given weight 1/(k-1). In this way every edge in G (provided neither end is connected to a vertex of degree '1') will have strength 2 in the line graph L(G) corresponding to the two ends that the edge has in G. It is straightforward to extend this definition of a weighted line graph to cases where the original graph G was directed or even weighted.[39] The principle in all cases is to ensure the line graph L(G) reflects the dynamics as well as the topology of the original graph G.

54.6.6 Line graphs of hypergraphs

Main article: Line graph of a hypergraph

The edges of a hypergraph may form an arbitrary family of sets, so the line graph of a hypergraph is the same as the intersection graph of the sets from the family.[25]

54.7 Notes

[1] Hemminger & Beineke (1978), p. 273.

[2] Harary (1972), p. 71.

[3] Whitney (1932); Krausz (1943); Harary (1972), Theorem 8.3, p. 72. Harary gives a simpliﬁed proof of this theorem by Jung (1966).

[4] Paschos, Vangelis Th. (2010), Combinatorial Optimization and Theoretical Computer Science: Interfaces and Perspectives, John Wiley & Sons, p. 394, ISBN 9780470393673, Clearly, there is a one-to-one correspondence between the matchings of a graph and the independent sets of its line graph.

[5] The need to consider isolated vertices when considering the connectivity of line graphs is pointed out by Cvetković, Rowl- inson & Simić (2004), p. 32.

[6] Harary (1972), Theorem 8.1, p. 72.

[7] Diestel, Reinhard (2006), Graph Theory, Graduate Texts in Mathematics 173, Springer, p. 112, ISBN 9783540261834. Also in free online edition, Chapter 5 (“Colouring”), p. 118.

[8] Lauri, Josef; Scapellato, Raﬀaele (2003), Topics in graph automorphisms and reconstruction, London Mathematical So- ciety Student Texts 54, Cambridge: Cambridge University Press, p. 44, ISBN 0-521-82151-7, MR 1971819. Lauri and Scapellato credit this result to Mark Watkins.

[9] Harary (1972), Theorem 8.8, p. 80.

[10] Ramezanpour, Karimipour & Mashaghi (2003).

[11] Jung (1966); Degiorgi & Simon (1995). 54.7. NOTES 195

[12] Zverovich (1997)

[13] van Dam, Edwin R.; Haemers, Willem H. (2003), “Which graphs are determined by their spectrum?", Linear Algebra and its Applications 373: 241–272, doi:10.1016/S0024-3795(03)00483-X, MR 2022290. See in particular Proposition 8, p. 262.

[14] Harary (1972), Theorem 8.6, p. 79. Harary credits this result to independent papers by L. C. Chang (1959) and A. J. Hoﬀman (1960).

[15] Chudnovsky, Maria; Robertson, Neil; Seymour, Paul; Thomas, Robin (2006), “The strong perfect graph theorem”, Annals of Mathematics 164 (1): 51–229, doi:10.4007/annals.2006.164.51. See also Roussel, F.; Rusu, I.; Thuillier, H. (2009), “The strong perfect graph conjecture: 40 years of attempts, and its resolution”, Discrete Mathematics 309 (20): 6092–6113, doi:10.1016/j.disc.2009.05.024, MR 2552645.

[16] Harary (1972), Theorem 8.7, p. 79. Harary credits this characterization of line graphs of complete bipartite graphs to Moon and Hoﬀman. The case of equal numbers of vertices on both sides had previously been proven by Shrikhande.

[17] Trotter (1977); de Werra (1978).

[18] Maﬀray (1992).

[19] Trotter (1977).

[20] Harary (1972), Theorem 8.4, p. 74, gives three equivalent characterizations of line graphs: the partition of the edges into cliques, the property of being claw-free and odd diamond-free, and the nine forbidden graphs of Beineke.

[21] Sumner, David P. (1974), “Graphs with 1-factors”, Proceedings of the American Mathematical Society (American Mathe- matical Society) 42 (1): 8–12, doi:10.2307/2039666, JSTOR 2039666, MR 0323648. Las Vergnas, M. (1975), “A note on matchings in graphs”, Cahiers du Centre d'Études de Recherche Opérationnelle 17 (2-3-4): 257–260, MR 0412042.

[22] Harary (1972), Theorem 8.5, p. 78. Harary credits the result to Gary Chartrand.

[23] Erdős, Paul; Saks, Michael; Sós, Vera T. (1986), “Maximum induced trees in graphs”, Journal of Combinatorial Theory, Series B 41 (1): 61–79, doi:10.1016/0095-8956(86)90028-6.

[24] Cvetković, Rowlinson & Simić (2004).

[25] Metelsky & Tyshkevich (1997)

[26] Harary (1972), Exercise 8.14, p. 83.

[27] This result is also Theorem 8.2 of Harary (1972).

[28] Harary (1972), Theorem 8.11, p. 81. Harary credits this result to Gary Chartrand.

[29] Sedláček (1964); Greenwell & Hemminger (1972).

[30] Archdeacon, Dan (1992), “The medial graph and voltage-current duality”, Discrete Mathematics 104 (2): 111–141, doi:10.1016/0012- 365X(92)90328-D, MR 1172842.

[31] McKee, T. A. (1989), “Graph-theoretic model of geographic duality”, Combinatorial Mathematics: Proceedings of the Third International Conference (New York, 1985), Ann. New York Acad. Sci. 555, New York: New York Acad. Sci., pp. 310–315, doi:10.1111/j.1749-6632.1989.tb22465.x, MR 1018637.

[32] Pugh, Anthony (1976), Polyhedra: A Visual Approach, University of California Press, ISBN 9780520030565.

[33] Loeb, Arthur Lee (1991), Space Structures—their Harmony and Counterpoint (5th ed.), Birkhäuser, ISBN 9783764335885.

[34] Weisstein, Eric W., “Rectiﬁcation”, MathWorld.

[35] Harary (1972), p. 82.

[36] Harary & Norman (1960).

[37] Zhang & Lin (1987).

[38] Evans & Lambiotte (2009).

[39] Evans & Lambiotte (2010). 196 CHAPTER 54. LINE GRAPH

54.8 References

• Beineke, L. W. (1968), “Derived graphs of digraphs”, in Sachs, H.; Voss, H.-J.; Walter, H.-J., Beiträge zur Graphentheorie, Leipzig: Teubner, pp. 17–33.

• Beineke, L. W. (1970), “Characterizations of derived graphs”, Journal of Combinatorial Theory 9 (2): 129– 135, doi:10.1016/S0021-9800(70)80019-9, MR 0262097.

• Cvetković, Dragoš; Rowlinson, Peter; Simić, Slobodan (2004), Spectral generalizations of line graphs, London Mathematical Society Lecture Note Series 314, Cambridge: Cambridge University Press, doi:10.1017/CBO9780511751752, ISBN 0-521-83663-8, MR 2120511.

• Degiorgi, Daniele Giorgio; Simon, Klaus (1995), “A dynamic algorithm for line graph recognition”, Graph- theoretic concepts in computer science (Aachen, 1995), Lecture Notes in Computer Science 1017, Berlin: Springer, pp. 37–48, doi:10.1007/3-540-60618-1_64, MR 1400011.

• Evans, T.S.; Lambiotte, R. (2009), “Line graphs, link partitions and overlapping communities”, Physical Review E 80: 016105, arXiv:0903.2181, doi:10.1103/PhysRevE.80.016105.

• Evans, T.S.; Lambiotte, R. (2010), “Line Graphs of Weighted Networks for Overlapping Communities”, European Physical Journal B 77: 265–272, arXiv:0912.4389, doi:10.1140/epjb/e2010-00261-8.

• Greenwell, D. L.; Hemminger, Robert L. (1972), “Forbidden subgraphs for graphs with planar line graphs”, Discrete Mathematics 2: 31–34, doi:10.1016/0012-365X(72)90058-1, MR 0297604.

• Harary, F.; Norman, R. Z. (1960), “Some properties of line digraphs”, Rendiconti del Circolo Matematico di Palermo 9 (2): 161–169, doi:10.1007/BF02854581.

• Harary, F. (1972), “8. Line Graphs”, Graph Theory (PDF), Massachusetts: Addison-Wesley, pp. 71–83.

• Hemminger, R. L.; Beineke, L. W. (1978), “Line graphs and line digraphs”, in Beineke, L. W.; Wilson, R. J., Selected Topics in Graph Theory, Academic Press Inc., pp. 271–305.

• Jung, H. A. (1966), “Zu einem Isomorphiesatz von H. Whitney für Graphen”, Mathematische Annalen (in German) 164: 270–271, doi:10.1007/BF01360250, MR 0197353.

• Krausz, J. (1943), “Démonstration nouvelle d'un théorème de Whitney sur les réseaux”, Mat. Fiz. Lapok 50: 75–85, MR 0018403.

• Lehot, Philippe G. H. (1974), “An optimal algorithm to detect a line graph and output its root graph”, Journal of the ACM 21: 569–575, doi:10.1145/321850.321853, MR 0347690.

• Maﬀray, Frédéric (1992), “Kernels in perfect line-graphs”, Journal of Combinatorial Theory, Series B 55 (1): 1–8, doi:10.1016/0095-8956(92)90028-V, MR 1159851.

• Metelsky, Yury; Tyshkevich, Regina (1997), “On line graphs of linear 3-uniform hypergraphs”, Journal of Graph Theory 25 (4): 243–251, doi:10.1002/(SICI)1097-0118(199708)25:4<243::AID-JGT1>3.0.CO;2-K.

• Ramezanpour, A.; Karimipour, V.; Mashaghi, A. (2003), “Generating correlated networks from uncorrelated ones”, Phys. Rev. E 67: 046107, doi:10.1103/physreve.67.046107.

• van Rooij, A. C. M.; Wilf, H. S. (1965), “The interchange graph of a ﬁnite graph”, Acta Mathematica Hungarica 16 (3–4): 263–269, doi:10.1007/BF01904834.

• Roussopoulos, N. D. (1973), “A max {m,n} algorithm for determining the graph H from its line graph G", Information Processing Letters 2 (4): 108–112, doi:10.1016/0020-0190(73)90029-X, MR 0424435.

• Sedláček, J. (1964), “Some properties of interchange graphs”, Theory of Graphs and its Applications (Proc. Sympos. Smolenice, 1963), Publ. House Czechoslovak Acad. Sci., Prague, pp. 145–150, MR 0173255.

• Sysło, Maciej M. (1982), “A labeling algorithm to recognize a line digraph and output its root graph”, Information Processing Letters 15 (1): 28–30, doi:10.1016/0020-0190(82)90080-1, MR 678028.

• Trotter, L. E., Jr. (1977), “Line perfect graphs”, Mathematical Programming 12 (2): 255–259, doi:10.1007/BF01593791, MR 0457293. 54.9. EXTERNAL LINKS 197

• de Werra, D. (1978), “On line perfect graphs”, Mathematical Programming 15 (2): 236–238, doi:10.1007/BF01609025, MR 509968. • Whitney, H. (1932), “Congruent graphs and the connectivity of graphs”, American Journal of Mathematics 54 (1): 150–168, doi:10.2307/2371086, JSTOR 2371086. • Zhang, Fu Ji; Lin, Guo Ning (1987), “On the de Bruijn–Good graphs”, Acta Math. Sinica 30 (2): 195–205, MR 0891925. • Зверович, И. Э. (1997), Аналог теоремы Уитни для реберных графов мультиграфов и реберные муль- тиграфы, Diskretnaya Matematika (in Russian) 9 (2): 98–105, doi:10.4213/dm478, MR 1468075. Translated into English as Zverovich, I. È. (1997), “An analogue of the Whitney theorem for edge graphs of multigraphs, and edge multigraphs”, Discrete Mathematics and Applications 7 (3): 287–294, doi:10.1515/dma.1997.7.3.287.

54.9 External links

• line graphs, Information System on Graph Class Inclusions External link in |publisher= (help) (last visited Sep 23 2013)

• Weisstein, Eric W., “Line Graph”, MathWorld. Chapter 55

Line graph of a hypergraph

The line graph of a hypergraph is the graph whose vertex set is the set of the hyperedges of the hypergraph, with two hyperedges adjacent when they have a nonempty intersection. In other words, the line graph of a hypergraph is the intersection graph of a family of ﬁnite sets. It is a generalization of the line graph of a graph. Questions about line graphs of hypergraphs are often generalizations of questions about line graphs of graphs. For instance, a hypergraph whose edges all have size k is called k -uniform. (A 2-uniform hypergraph is a graph.). In hypergraph theory, it is often natural to require that hypergraphs be k-uniform. Every graph is the line graph of some hypergraph, but, given a ﬁxed edge size k, not every graph is a line graph of some k-uniform hypergraph. A main problem is to characterize those that are, for each k ≥ 3. A hypergraph is linear if each pair of hyperedges intersects in at most one vertex. Every graph is the line graph, not only of some hypergraph, but of some linear hypergraph (Berge 1989).

55.1 Line graphs of k-uniform hypergraphs, k ≥ 3

Beineke (1968) characterized line graphs of graphs by a list of 9 forbidden induced subgraphs. (See the article on line graphs.) No characterization by forbidden induced subgraphs is known of line graphs of k-uniform hypergraphs for any k ≥ 3, and Lovász (1977) showed there is no such characterization by a ﬁnite list if k = 3. Krausz (1943) characterized line graphs of graphs in terms of clique covers. (See Line Graphs.) A global characterization of Krausz type for the line graphs of k-uniform hypergraphs for any k ≥ 3 was given by Berge (1989).

55.2 Line graphs of k-uniform linear hypergraphs, k ≥ 3

A global characterization of Krausz type for the line graphs of k-uniform linear hypergraphs for any k ≥ 3 was given by Naik et al. (1980). At the same time, they found a finite list of forbidden induced subgraphs for linear 3-uniform hypergraphs with minimum vertex degree at least 69. Metelsky & Tyshkevich (1997) and Jacobson, Kézdy & Lehel (1997) improved this bound to 19. At last Skums, Suzdal' & Tyshkevich (2005) reduced this bound to 16. Metelsky & Tyshkevich (1997) also proved that, if k > 3, no such finite list exists for linear k-uniform hypergraphs, no matter what lower bound is placed on the degree. The difficulty in finding a characterization of linear k-uniform hypergraphs is due to the fact that there are infinitely many forbidden induced subgraphs. To give examples, for m > 0, consider a chain of m diamond graphs such that the consecutive diamonds share vertices of degree two. For k ≥ 3, add pendant edges at every vertex of degree 2 or 4 to get one of the families of minimal forbidden subgraphs of Naik, Rao, and Shrikhande et al. (1980, 1982) as shown here. This does not rule out either the existence of a polynomial recognition or the possibility of a forbidden induced subgraph characterization similar to Beineke’s of line graphs of graphs. There are some interesting characterizations available for line graphs of linear k-uniform hypergraphs due to various authors (Naik, Rao & Shrikhande et al. 1980, 1982, Jacobson, Kézdy & Lehel 1997, Metelsky & Tyshkevich 1997, and Zverovich 2004) under constraints on the minimum degree or the minimum edge degree of G. Minimum edge degree at least k3−2k2+1 in Naik et al. (1980) is reduced to 2k2−3k+1 in Jacobson, Kézdy & Lehel (1997) and

198 55.3. REFERENCES 199

Zverovich (2004) to characterize line graphs of k-uniform linear hypergraphs for any k ≥ 3. The complexity of recognizing line graphs of linear k-uniform hypergraphs without any constraint on minimum degree (or minimum edge-degree) is not known. For k = 3 and minimum degree at least 19, recognition is possible in polynomial time (Jacobson, Kézdy & Lehel 1997 and Metelsky & Tyshkevich 1997). Skums, Suzdal' & Tyshkevich (2005) reduced the minimum degree to 10. There are many interesting open problems and conjectures in Naik et al., Jacoboson et al., Metelsky et al. and Zverovich.

55.3 References

• Beineke, L. W. (1968), “On derived graphs and digraphs”, in Sachs, H.; Voss, H.; Walther, H., Beitrage zur Graphentheorie, Leipzig: Teubner, pp. 17–23.

• Berge, C. (1989), Hypergraphs: Combinatorics of Finite Sets, Amsterdam: North-Holland, MR 1013569. Translated from the French.

• Bermond, J. C.; Heydemann, M. C.; Sotteau, D. (1977), “Line graphs of hypergraphs I”, Discrete Mathematics 18: 235–241, doi:10.1016/0012-365X(77)90127-3, MR 0463003.

• Heydemann, M. C.; Sotteau, D. (1976), “Line graphs of hypergraphs II”, Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely, 1976), Colloq. Math. Soc. J. Bolyai 18, pp. 567–582, MR 0519291.

• Krausz, J. (1943), “Démonstration nouvelle d'une théorème de Whitney sur les réseaux”, Mat. Fiz. Lapok 50: 75–85, MR 0018403. (In Hungarian, with French abstract.)

• Lovász, L. (1977), “Problem 9”, Beitrage zur Graphentheorie und deren Ansendungen, Vortgetragen auf dem international Colloquium in Oberhof (DDR), p. 313.

• Jacobson, M. S.; Kézdy, Andre E.; Lehel, Jeno (1997), “Recognizing intersection graphs of linear uniform hypergraphs”, Graphs and Combinatorics 13: 359–367, doi:10.1007/BF03353014, MR 1485929.

• Metelsky, Yury; Tyshkevich, Regina (1997), “On line graphs of linear 3-uniform hypergraphs”, Journal of Graph Theory 25: 243–251, doi:10.1002/(SICI)1097-0118(199708)25:4<243::AID-JGT1>3.0.CO;2-K, MR 1459889.

• Naik, Ranjan N.; Rao, S. B.; Shrikhande, S. S.; Singhi, N. M. (1980), “Intersection graphs of k-uniform hypergraphs”, Combinatorial mathematics, optimal designs and their applications (Proc. Sympos. Combin. Math. and Optimal Design, Colorado State Univ., Fort Collins, Colo., 1978), Annals of Discrete Mathematics 6, pp. 275–279, MR 0593539.

• Naik, Ranjan N.; Rao, S. B.; Shrikhande, S. S.; Singhi, N. M. (1982), “Intersection graphs of k-uniform linear hypergraphs”, European J. Combinatorics 3: 159–172, doi:10.1016/s0195-6698(82)80029-2, MR 0670849.

• Skums, P. V.; Suzdal', S. V.; Tyshkevich, R. I. (2009), “Edge intersection of linear 3-uniform hypergraphs”, Discrete Mathematics 309: 3500–3517, doi:10.1016/j.disc.2007.12.082. 200 CHAPTER 55. LINE GRAPH OF A HYPERGRAPH

• Zverovich, Igor E. (2004), “A solution to a problem of Jacobson, Kézdy and Lehel”, Graphs and Combinatorics 20 (4): 571–577, doi:10.1007/s00373-004-0572-1, MR 2108401.

• Voloshin, Vitaly I. (2009), Introduction to Graph and Hypergraph Theory, New York: Nova Science Publishers, Inc., MR 2514872 Chapter 56

List of graphs

This partial list of graphs contains deﬁnitions of graphs and graph families which are known by particular names, but do not have a Wikipedia article of their own. For collected deﬁnitions of graph theory terms that do not refer to individual graph types, such as vertex and path, see Glossary of graph theory. For links to existing articles about particular kinds of graphs, see Category:Graphs.

56.1 Gear

A gear graph, denoted Gn is a graph obtained by inserting an extra vertex between each pair of adjacent vertices on the perimeter of a wheel graph Wn. Thus, Gn has 2n+1 vertices and 3n edges.[1] Gear graphs are examples of squaregraphs, and play a key role in the forbidden graph characterization of squaregraphs.[2] Gear graphs are also known as cogwheels and bipartite wheels.

56.2 Grid

A grid graph is a unit distance graph corresponding to the square lattice, so that it is isomorphic to the graph having a vertex corresponding to every pair of integers (a, b), and an edge connecting (a, b) to (a+1, b) and (a, b+1). The ﬁnite grid graph Gm,n is an m×n rectangular graph isomorphic to the one obtained by restricting the ordered pairs to the range 0 ≤ a < m, 0 ≤ b < n. Grid graphs can be obtained as the Cartesian product of two paths: Gm,n = Pm × Pn. Every grid graph is a median graph.[3]

56.3 Helm

A helm graph, denoted Hn is a graph obtained by attaching a single edge and node to each node of the outer circuit of a wheel graph Wn.[4][5]

56.4 Lobster

A lobster graph is a tree in which all the vertices are within distance 2 of a central path.[6][7] Compare caterpillar.

56.5 Web

The web graph Wn,r is a graph consisting of r concentric copies of the cycle graph Cn, with corresponding vertices connected by “spokes”. Thus Wn,₁ is the same graph as Cn, and W,₂ is a prism. A web graph has also been deﬁned as a prism graph Yn₊₁, ₃, with the edges of the outer cycle removed.[5][8]

201 202 CHAPTER 56. LIST OF GRAPHS

The web graph W4,2 is a cube.

56.6 See also

Gallery of named graphs

56.7 References

[1] Weisstein, Eric W., “Gear graph”, MathWorld.

[2] Bandelt, H.-J.; Chepoi, V.; Eppstein, D. (2010), “Combinatorics and geometry of ﬁnite and inﬁnite squaregraphs”, SIAM Journal on Discrete Mathematics 24 (4): 1399–1440, arXiv:0905.4537, doi:10.1137/090760301

[3] Weisstein, Eric W., “Grid graph”, MathWorld.

[4] Weisstein, Eric W., “Helm graph”, MathWorld.

[5] http://www.combinatorics.org/Surveys/ds6.pdf

[6] “Google Discussiegroepen”. Groups.google.com. Retrieved 2014-02-05.

[7] Weisstein, Eric W., “Lobster”, MathWorld.

[8] Weisstein, Eric W., “Web graph”, MathWorld. Chapter 57

Lévy family of graphs

In graph theory, a branch of mathematics, a Lévy family of graphs is a family of graphs Gn, n = 1, 2, 3, ..., which possess a certain type of “compactness” or “tangledness”. Many naturally occurring families of graphs are Lévy families. Many mathematicians have noted this fact and have expressed surprise that it does not appear to have a ready explanation. Formally, a family of graphs Gn, n = 1, 2, 3, ..., is a Lévy family if, for any ε > 0

lim α (Gn, ε) = 0 n−→∞ where

{ } A α(G, ε) = max 1 − (εD) : A ⊆ G, |A| > |G|/2 . |G|

Here D is the graph diameter of G, and A₍n₎ is the n-graph neighborhood of A. Note that the maximization ranges over subsets A of G, subject to A being over half the size of G In words, this means that one can take a subset of size at least half of G, and blow it up by only ϵ of the graph diameter, and end up with nearly all the set. Long “stringy” (i.e. not “compact”) families of graphs such as the cycle graph of order n clearly don't have such a property: one could consider a subset comprising the n/2 neighborhood of a point (midnight to six o'clock, say). The graph has graph diameter D of about n/2. So the ϵD -neighborhood of the subset is only of size about n/2. A Levy family would have this neighborhood covering almost all the set. It should be clear that a Levy family must have a very special type of compact structure.

• Hypercube graphs of order n are known to be a Lévy family. • If Sn is the graph with points that are elements of the permutation group of n elements, with edges joining points that diﬀer by a transposition, then the series Si, i=1,2,..., is a Lévy family.

57.1 References

• Bollobás (editor). Probabilistic combinatorics and its applications. American Mathematical Society, 1991 (p63)

203 Chapter 58

Medial graph

A plane graph (in blue) and its medial graph (in red).

In the mathematical discipline of graph theory, the medial graph of plane graph G is another graph M(G) that represents the adjacencies between edges in the faces of G. Medial graphs were introduced in 1922 by Ernst Steinitz to study combinatorial properties of convex polyhedra,[1] although the inverse construction was already used by Peter Tait in 1877 in his foundational study of knots and links.[2][3]

58.1 Formal deﬁnition

Given a connected plane graph G, its medial graph M(G) has

• a vertex for each edge of G and

• an edge between two vertices for each face of G in which their corresponding edges occur consecutively.

The medial graph of a disconnected graph is the disjoint union of the medial graphs of each connected component. The deﬁnition of medial graph also extends without modiﬁcation to graph embeddings on surfaces of higher genus.

58.2 Properties

• The medial graph of any plane graph is a 4-regular plane graph.

204 58.3. APPLICATIONS 205

The two red graphs are both medial graphs of the blue graph, but they are not isomorphic.

• For any plane graph G, the medial graph of G and the medial graph of the dual graph of G are isomorphic. Conversely, for any 4-regular plane graph H, the only two plane graphs with medial graph H are dual to each other.[4]

• Since the medial graph depends on a particular embedding, the medial graph of a planar graph is not unique; the same planar graph can have non-isomorphic medial graphs. In the picture, the red graphs are not isomorphic because the two vertices with self loops share an edge in one graph but not in the other.

• Every 4-regular plane graph is the medial graph of some plane graph. For a connected 4-regular plane graph H, a planar graph G with H as its medial graph can be constructed as follows. Color the faces of H with just two colors, which is possible since H is Eulerian (and thus the dual graph of H is bipartite). The vertices in G correspond to the faces of a single color in H. These vertices are connected by an edge for each vertex shared by their corresponding faces in H. Note that performing this construction using the faces of the other color as the vertices produces the dual graph of G.

58.3 Applications

For a plane graph G, twice the evaluation of the Tutte polynomial at the point (3,3) equals the sum over weighted Eulerian orientations in the medial graph of G, where the weight of an orientation is 2 to the number of saddle vertices of the orientation (that is, the number of vertices with incident edges cyclically ordered “in, out, in out”).[5] Since the Tutte polynomial is invariant under embeddings, this result shows that every medial graph has the same sum of these weighted Eulerian orientations. 206 CHAPTER 58. MEDIAL GRAPH

58.4 Directed medial graph

A plane graph (in blue) and its directed medial graph (in red).

The medial graph deﬁnition can be extended to include an orientation. First, the faces of the medial graph are colored black if they contain a vertex of the original graph and white otherwise. This coloring causes each edge of the medial graph to be bordered by one black face and one white face. Then each edge is oriented so that the black face is on its left. A plane graph and its dual do not have the same directed medial graph; their directed medial graphs are the transpose of each other. Using the directed medial graph, one can eﬀectively generalize the result on evaluations of the Tutte polynomial at (3,3). For a plane graph G, n times the evaluation of the Tutte polynomial at the point (n+1,n+1) equals the weighted sum over all edge colorings using n colors in the directed medial graph of G so that each (possibly empty) set of monochromatic edges forms a directed Eulerian graph, where the weight of a directed Eulerian orientation is 2 to the number of monochromatic vertices.[6]

58.5 See also

• Line graph • Knots and graphs • Tait graph • Rectiﬁcation (geometry) - The equivalent operation on polyhedrons

58.6 References

[1] Steinitz, Ernst (1922), “Polyeder und Raumeinteilungen”, Encyclopädie der mathematischen Wissenschaften, Band 3 (Ge- ometries), pp. 1–139

[2] Tait, Peter G. (1876–1877). “On Knots I”. Proceedings of the Royal Society of Edinburgh 28: 145–190. Revised May 11, 1877.

[3] Tait, Peter G. (1876–1877). “On Links (Abstract)". Proceedings of the Royal Society of Edinburgh 9 (98): 321–332.

[4] Gross, Jonathan L.; Yellen, Jay, eds. (2003). Handbook of Graph Theory. CRC Press. p. 724. ISBN 978-1584880905.

[5] Las Vergnas, Michel (1988), “On the evaluation at (3, 3) of the Tutte polynomial of a graph”, Journal of Combinatorial Theory, Series B 35 (3): 367–372, doi:10.1016/0095-8956(88)90079-2, ISSN 0095-8956

[6] Ellis-Monaghan, Joanna A. (2004). “Identities for circuit partition polynomials, with applications to the Tutte polynomial”. Advances in Applied Mathematics 32 (1-2): 188–197. doi:10.1016/S0196-8858(03)00079-4. ISSN 0196-8858.

• Brylawski, Thomas; Oxley, James (1992). “The Tutte Polynomial and Its Applications”. In White, Neil. Matriod Applications (PDF). Cambridge University Press. pp. 123–225. Chapter 59

Median graph

a b

c m(a,b,c)

The median of three vertices in a median graph

In graph theory, a division of mathematics, a median graph is an undirected graph in which every three vertices a, b, and c have a unique median: a vertex m(a,b,c) that belongs to shortest paths between each pair of a, b, and c. The concept of median graphs has long been studied, for instance by Birkhoﬀ & Kiss (1947) or (more explicitly) by Avann (1961), but the ﬁrst paper to call them “median graphs” appears to be Nebeský (1971). As Chung, Graham, and Saks write, “median graphs arise naturally in the study of ordered sets and discrete distributive lattices, and have an extensive literature”.[1] In phylogenetics, the Buneman graph representing all maximum parsimony evolutionary trees is a median graph.[2] Median graphs also arise in social choice theory: if a set of alternatives has the structure of a median graph, it is possible to derive in an unambiguous way a majority preference among them.[3] Additional surveys of median graphs are given by Klavžar & Mulder (1999), Bandelt & Chepoi (2008), and Knuth (2008).

207 208 CHAPTER 59. MEDIAN GRAPH

59.1 Examples a

m(a,b,c)

The median of three vertices in a tree, showing the subtree formed by the union of shortest paths between the vertices.

Every tree is a median graph.[4] To see this, observe that in a tree, the union of the three shortest paths between pairs 59.1. EXAMPLES 209 of the three vertices a, b, and c is either itself a path, or a subtree formed by three paths meeting at a single central node with degree three. If the union of the three paths is itself a path, the median m(a,b,c) is equal to one of a, b, or c, whichever of these three vertices is between the other two in the path. If the subtree formed by the union of the three paths is not a path, the median of the three vertices is the central degree-three node of the subtree. Additional examples of median graphs are provided by the grid graphs. In a grid graph, the coordinates of the median m(a,b,c) can be found as the median of the coordinates of a, b, and c. Conversely, it turns out that, in every median graph, one may label the vertices by points in an integer lattice in such a way that medians can be calculated coordinatewise in this way.[5]

A squaregraph.

Squaregraphs, planar graphs in which all interior faces are quadrilaterals and all interior vertices have four or more incident edges, are another subclass of the median graphs.[6] A polyomino is a special case of a squaregraph and therefore also forms a median graph. The simplex graph κ(G) of an arbitrary undirected graph G has a node for every clique (complete subgraph) of G; two nodes are linked by an edge if the corresponding cliques diﬀer by one vertex. The median of a given triple of cliques may be formed by using the majority rule to determine which vertices of the cliques to include; the simplex graph is a median graph in which this rule determines the median of each triple of vertices. No cycle graph of length other than four can be a median graph, because every such cycle has three vertices a, b, and c such that the three shortest paths wrap all the way around the cycle without having a common intersection. For 210 CHAPTER 59. MEDIAN GRAPH

such a triple of vertices, there can be no median.

59.2 Equivalent deﬁnitions

In an arbitrary graph, for each two vertices a and b, the minimal number of edges between them is called their distance, denoted by d(x,y). The interval of vertices that lie on shortest paths between a and b is deﬁned as

I(a,b) = {v | d(a,b) = d(a,v) + d(v,b)}.

A median graph is deﬁned by the property that, for every three vertices a, b, and c, these intervals intersect in a single point:

For all a, b, and c, |I(a,b) ∩ I(a,c) ∩ I(b,c)| = 1.

Equivalently, for every three vertices a, b, and c one can ﬁnd a vertex m(a,b,c) such that the unweighted distances in the graph satisfy the equalities

• d(a,b) = d(a,m(a,b,c)) + d(m(a,b,c),b) • d(a,c) = d(a,m(a,b,c)) + d(m(a,b,c),c) • d(b,c) = d(b,m(a,b,c)) + d(m(a,b,c),c)

and m(a,b,c) is the only vertex for which this is true. It is also possible to define median graphs as the solution sets of 2-satisfiability problems, as the retracts of hypercubes, as the graphs of finite median algebras, as the Buneman graphs of Helly split systems, and as the graphs of windex 2; see the sections below.

59.3 Distributive lattices and median algebras

In lattice theory, the graph of a ﬁnite lattice has a vertex for each lattice element and an edge for each pair of elements in the covering relation of the lattice. Lattices are commonly presented visually via Hasse diagrams, which are drawings of graphs of lattices. These graphs, especially in the case of distributive lattices, turn out to be closely related to median graphs. In a distributive lattice, Birkhoﬀ’s self-dual ternary median operation[7]

m(a,b,c) = (a ∧ b) ∨ (a ∧ c) ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c) ∧ (b ∨ c),

satisﬁes certain key axioms, which it shares with the usual median of numbers in the range from 0 to 1 and with median algebras more generally:

• Idempotence: m(a,a,b) = a for all a and b. • Commutativity: m(a,b,c) = m(a,c,b) = m(b,a,c) = m(b,c,a) = m(c,a,b) = m(c,b,a) for all a, b, and c. • Distributivity: m(a,m(b,c,d),e) = m(m(a,b,e),c,m(a,d,e)) for all a, b, c, d, and e. • Identity elements: m(0,a,1) = a for all a.

The distributive law may be replaced by an associative law:[8]

• Associativity: m(x,w,m(y,w,z)) = m(m(x,w,y),w,z)

The median operation may also be used to deﬁne a notion of intervals for distributive lattices: 59.4. CONVEX SETS AND HELLY FAMILIES 211

I(a,b) = {x | m(a,x,b) = x} = {x | a ∧ b ≤ x ≤ a ∨ b}.[9]

The graph of a finite distributive lattice has an edge between vertices a and b whenever I(a,b) = {a,b}. For every two vertices a and b of this graph, the interval I(a,b) defined in lattice-theoretic terms above consists of the vertices on shortest paths from a to b, and thus coincides with the graph-theoretic intervals defined earlier. For every three lattice elements a, b, and c, m(a,b,c) is the unique intersection of the three intervals I(a,b), I(a,c), and I(b,c).[10] Therefore, the graph of an arbitrary finite distributive lattice is a median graph. Conversely, if a median graph G contains two vertices 0 and 1 such that every other vertex lies on a shortest path between the two (equivalently, m(0,a,1) = a for all a), then we may define a distributive lattice in which a ∧ b = m(a,0,b) and a ∨ b = m(a,1,b), and G will be the graph of this lattice.[11] Duffus & Rival (1983) characterize graphs of distributive lattices directly as diameter-preserving retracts of hypercubes. More generally, every median graph gives rise to a ternary operation m satisfying idempotence, commutativity, and distributivity, but possibly without the identity elements of a distributive lattice. Every ternary operation on a finite set that satisfies these three properties (but that does not necessarily have 0 and 1 elements) gives rise in the same way to a median graph.[12]

59.4 Convex sets and Helly families

In a median graph, a set S of vertices is said to be convex if, for every two vertices a and b belonging to S, the whole interval I(a,b) is a subset of S. Equivalently, given the two deﬁnitions of intervals above, S is convex if it contains every shortest path between two of its vertices, or if it contains the median of every set of three points at least two of which are from S. Observe that the intersection of every pair of convex sets is itself convex.[13] The convex sets in a median graph have the Helly property: if F is an arbitrary family of pairwise-intersecting convex sets, then all sets in F have a common intersection.[14] For, if F has only three convex sets S, T, and U in it, with a in the intersection of the pair S and T, b in the intersection of the pair T and U, and c in the intersection of the pair S and U, then every shortest path from a to b must lie within T by convexity, and similarly every shortest path between the other two pairs of vertices must lie within the other two sets; but m(a,b,c) belongs to paths between all three pairs of vertices, so it lies within all three sets, and forms part of their common intersection. If F has more than three convex sets in it, the result follows by induction on the number of sets, for one may replace an arbitrary pair of sets in F by their intersection, using the result for triples of sets to show that the replaced family is still pairwise intersecting. A particularly important family of convex sets in a median graph, playing a role similar to that of halfspaces in Euclidean space, are the sets

Wuv = {w | d(w,u) < d(w,v)}

defined for each edge uv of the graph. In words, Wuv consists of the vertices closer to u than to v, or equivalently the vertices w such that some shortest path from v to w goes through u. To show that Wuv is convex, let w1w2...wk be an arbitrary shortest path that starts and ends within Wuv; then w2 must also lie within Wuv, for otherwise the two points m1 = m(u,w1,wk) and m2 = m(m1,w2...wk) could be shown (by considering the possible distances between the vertices) to be distinct medians of u, w1, and wk, contradicting the definition of a median graph which requires medians to be unique. Thus, each successive vertex on a shortest path between two vertices of Wuv also lies within Wuv, so Wuv contains all shortest paths between its nodes, one of the definitions of convexity. The Helly property for the sets Wuv plays a key role in the characterization of median graphs as the solution of 2-satisfiability instances, below.

59.5 2-satisﬁability

Median graphs have a close connection to the solution sets of 2-satisﬁability problems that can be used both to characterize these graphs and to relate them to adjacency-preserving maps of hypercubes.[15] A 2-satisﬁability instance consists of a collection of Boolean variables and a collection of clauses, constraints on certain pairs of variables requiring those two variables to avoid certain combinations of values. Usually such problems are expressed in conjunctive normal form, in which each clause is expressed as a disjunction and the whole set of constraints is expressed as a conjunction of clauses, such as 212 CHAPTER 59. MEDIAN GRAPH

(x11 ∨ x12) ∧ (x21 ∨ x22) ∧ · · · ∧ (xn1 ∨ xn2) ∧ · · · .

A solution to such an instance is an assignment of truth values to the variables that satisfies all the clauses, or equivalently that causes the conjunctive normal form expression for the instance to become true when the variable values are substituted into it. The family of all solutions has a natural structure as a median algebra, where the median of three solutions is formed by choosing each truth value to be the majority function of the values in the three solutions; it is straightforward to verify that this median solution cannot violate any of the clauses. Thus, these solutions form a median graph, in which the neighbor of each solution is formed by negating a set of variables that are all constrained to be equal or unequal to each other. Conversely, every median graph G may be represented in this way as the solution set to a 2-satisfiability instance. To find such a representation, create a 2-satisfiability instance in which each variable describes the orientation of one of the edges in the graph (an assignment of a direction to the edge causing the graph to become directed rather than undirected) and each constraint allows two edges to share a pair of orientations only when there exists a vertex v such that both orientations lie along shortest paths from other vertices to v. Each vertex v of G corresponds to a solution to this 2-satisfiability instance in which all edges are directed towards v. Each solution to the instance must come from some vertex v in this way, where v is the common intersection of the sets Wuw for edges directed from w to u; this common intersection exists due to the Helly property of the sets Wuw. Therefore, the solutions to this 2-satisfiability instance correspond one-for-one with the vertices of G.

59.6 Retracts of hypercubes

A retraction of a graph G is an adjacency-preserving map from G to one of its subgraphs.[16] More precisely, it is graph homomorphism φ from G to itself such that φ(v) = v for each vertex v in the subgraph φ(G). The image of the retraction is called a retract of G. Retractions are examples of metric maps: the distance between φ(v) and φ(w), for every v and w, is at most equal to the distance between v and w, and is equal whenever v and w both belong to φ(G). Therefore, a retract must be an isometric subgraph of G: distances in the retract equal those in G. If G is a median graph, and a, b, and c are an arbitrary three vertices of a retract φ(G), then φ(m(a,b,c)) must be a median of a, b, and c, and so must equal m(a,b,c). Therefore, φ(G) contains medians of all triples of its vertices, and must also be a median graph. In other words, the family of median graphs is closed under the retraction operation.[17] A hypercube graph, in which the vertices correspond to all possible k-bit bitvectors and in which two vertices are adjacent when the corresponding bitvectors differ in only a single bit, is a special case of a k-dimensional grid graph and is therefore a median graph. The median of three bitvectors a, b, and c may be calculated by computing, in each bit position, the majority function of the bits of a, b, and c. Since median graphs are closed under retraction, and include the hypercubes, every retract of a hypercube is a median graph. Conversely, every median graph must be the retract of a hypercube.[18] This may be seen from the connection, described above, between median graphs and 2-satisfiability: let G be the graph of solutions to a 2-satisfiability instance; without loss of generality this instance can be formulated in such a way that no two variables are always equal or always unequal in every solution. Then the space of all truth assignments to the variables of this instance forms a hypercube. For each clause, formed as the disjunction of two variables or their complements, in the 2- satisfiability instance, one can form a retraction of the hypercube in which truth assignments violating this clause are mapped to truth assignments in which both variables satisfy the clause, without changing the other variables in the truth assignment. The composition of the retractions formed in this way for each of the clauses gives a retraction of the hypercube onto the solution space of the instance, and therefore gives a representation of G as the retract of a hypercube. In particular, median graphs are isometric subgraphs of hypercubes, and are therefore partial cubes. However, not all partial cubes are median graphs; for instance, a six-vertex cycle graph is a partial cube but is not a median graph. As Imrich & Klavžar (2000) describe, an isometric embedding of a median graph into a hypercube may be constructed in time O(m log n), where n and m are the numbers of vertices and edges of the graph respectively.[19] 59.7. TRIANGLE-FREE GRAPHS AND RECOGNITION ALGORITHMS 213

59.7 Triangle-free graphs and recognition algorithms

The problems of testing whether a graph is a median graph, and whether a graph is triangle-free, both had been well studied when Imrich, Klavžar & Mulder (1999) observed that, in some sense, they are computationally equivalent.[20] Therefore, the best known time bound for testing whether a graph is triangle-free, O(m1.41),[21] applies as well to testing whether a graph is a median graph, and any improvement in median graph testing algorithms would also lead to an improvement in algorithms for detecting triangles in graphs. In one direction, suppose one is given as input a graph G, and must test whether G is triangle-free. From G, construct a new graph H having as vertices each set of zero, one, or two adjacent vertices of G. Two such sets are adjacent in H when they differ by exactly one vertex. An equivalent description of H is that it is formed by splitting each edge of G into a path of two edges, and adding a new vertex connected to all the original vertices of G. This graph H is by construction a partial cube, but it is a median graph only when G is triangle-free: if a, b, and c form a triangle in G, then {a,b}, {a,c}, and {b,c} have no median in H, for such a median would have to correspond to the set {a,b,c}, but sets of three or more vertices of G do not form vertices in H. Therefore, G is triangle-free if and only if H is a median graph. In the case that G is triangle-free, H is its simplex graph. An algorithm to test efficiently whether H is a median graph could by this construction also be used to test whether G is triangle-free. This transformation preserves the computational complexity of the problem, for the size of H is proportional to that of G. The reduction in the other direction, from triangle detection to median graph testing, is more involved and depends on the previous median graph recognition algorithm of Hagauer, Imrich & Klavžar (1999), which tests several necessary conditions for median graphs in near-linear time. The key new step involves using a breadth first search to partition the graph into levels according to their distances from some arbitrarily chosen root vertex, forming a graph in each level in which two vertices are adjacent if they share a common neighbor in the previous level, and searching for triangles in these graphs. The median of any such triangle must be a common neighbor of the three triangle vertices; if this common neighbor does not exist, the graph is not a median graph. If all triangles found in this way have medians, and the previous algorithm finds that the graph satisfies all the other conditions for being a median graph, then it must actually be a median graph. Note that this algorithm requires, not just the ability to test whether a triangle exists, but a list of all triangles in the level graph. In arbitrary graphs, listing all triangles sometimes requires Ω(m3/2) time, as some graphs have that many triangles, however Hagauer et al. show that the number of triangles arising in the level graphs of their reduction is near-linear, allowing the Alon et al. fast matrix multiplication based technique for finding triangles to be used.

59.8 Evolutionary trees, Buneman graphs, and Helly split systems

Phylogeny is the inference of evolutionary trees from observed characteristics of species; such a tree must place the species at distinct vertices, and may have additional latent vertices, but the latent vertices are required to have three or more incident edges and must also be labeled with characteristics. A characteristic is binary when it has only two possible values, and a set of species and their characteristics exhibit perfect phylogeny when there exists an evolutionary tree in which the vertices (species and latent vertices) labeled with any particular characteristic value form a contiguous subtree. If a tree with perfect phylogeny is not possible, it is often desired to find one exhibiting maximum parsimony, or equivalently, minimizing the number of times the endpoints of a tree edge have different values for one of the characteristics, summed over all edges and all characteristics. Buneman (1971) described a method for inferring perfect phylogenies for binary characteristics, when they exist. His method generalizes naturally to the construction of a median graph for any set of species and binary characteristics, which has been called the median network or Buneman graph[22] and is a type of phylogenetic network. Every maximum parsimony evolutionary tree embeds into the Buneman graph, in the sense that tree edges follow paths in the graph and the number of characteristic value changes on the tree edge is the same as the number in the corresponding path. The Buneman graph will be a tree if and only if a perfect phylogeny exists; this happens when there are no two incompatible characteristics for which all four combinations of characteristic values are observed. To form the Buneman graph for a set of species and characteristics, first, eliminate redundant species that are indistin- guishable from some other species and redundant characteristics that are always the same as some other characteristic. Then, form a latent vertex for every combination of characteristic values such that every two of the values exist in some known species. In the example shown, there are small brown tailless mice, small silver tailless mice, small brown tailed mice, large brown tailed mice, and large silver tailed mice; the Buneman graph method would form a latent vertex corresponding to an unknown species of small silver tailed mice, because every pairwise combination (small and silver, small and tailed, and silver and tailed) is observed in some other known species. However, the 214 CHAPTER 59. MEDIAN GRAPH method would not infer the existence of large brown tailless mice, because no mice are known to have both the large and tailless traits. Once the latent vertices are determined, form an edge between every pair of species or latent vertices that differ in a single characteristic. One can equivalently describe a collection of binary characteristics as a split system, a family of sets having the property that the complement set of each set in the family is also in the family. This split system has a set for each characteristic value, consisting of the species that have that value. When the latent vertices are included, the resulting split system has the Helly property: every pairwise intersecting subfamily has a common intersection. In some sense median graphs are characterized as coming from Helly split systems: the pairs (Wuv, Wvu) defined for each edge uv of a median graph form a Helly split system, so if one applies the Buneman graph construction to this system no latent vertices will be needed and the result will be the same as the starting graph.[23] Bandelt et al. (1995) and Bandelt, Macaulay & Richards (2000) describe techniques for simplified hand calculation of the Buneman graph, and use this construction to visualize human genetic relationships.

59.9 Additional properties

• The Cartesian product of every two median graphs is another median graph. Medians in the product graph may be computed by independently finding the medians in the two factors, just as medians in grid graphs may be computed by independently finding the median in each linear dimension. • The windex of a graph measures the amount of lookahead needed to optimally solve a problem in which one is given a sequence of graph vertices si, and must find as output another sequence of vertices ti minimizing the sum of the distances d(si,ti) and d(ti ₋ ₁,ti). Median graphs are exactly the graphs that have windex 2. In a median graph, the optimal choice is to set ti = m(ti ₋ ₁,si,si ₊ ₁).[1] • The property of having a unique median is also called the unique Steiner point property.[1] An optimal Steiner tree for three vertices a, b, and c in a median graph may be found as the union of three shortest paths, from a, b, and c to m(a,b,c). Bandelt & Barthélémy (1984) study more generally the problem of finding the vertex minimizing the sum of distances to each of a given set of vertices, and show that it has a unique solution for any odd number of vertices in a median graph. They also show that this median of a set S of vertices in a median graph satisfies the Condorcet criterion for the winner of an election: compared to any other vertex, it is closer to a majority of the vertices in S.

• As with partial cubes more generally, every median graph with n vertices has at most (n/2) log2 n edges. However, the number of edges cannot be too small: Klavžar, Mulder & Škrekovski (1998) prove that in every median graph the inequality 2n − m − k ≤ 2 holds, where m is the number of edges and k is the dimension of the hypercube that the graph is a retract of. This inequality is an equality if and only if the median graph contains no cubes. This is a consequence of another identity for median graphs: the Euler characteristic Σ(−1)dim(Q) is always equal to one, where the sum is taken over all hypercube subgraphs Q of the given median graph.[24] • The only regular median graphs are the hypercubes.[25]

59.10 Notes

[1] Chung, Graham & Saks (1987).

[2] Buneman (1971); Dress et al. (1997); Dress, Huber & Moulton (1997).

[3] Bandelt & Barthélémy (1984); Day & McMorris (2003).

[4] Imrich & Klavžar (2000), Proposition 1.26, p. 24.

[5] This follows immediately from the characterization of median graphs as retracts of hypercubes, described below.

[6] Soltan, Zambitskii & Prisăcaru (1973); Chepoi, Dragan & Vaxès (2002); Chepoi, Fanciullini & Vaxès (2004).

[7] Birkhoff & Kiss (1947) credit the definition of this operation to Birkhoff, G. (1940), Lattice Theory, American Mathe- matical Society, p. 74.

[8] Knuth (2008), p. 65, and exercises 75 and 76 on pp. 89–90. Knuth states that a simple proof that associativity implies distributivity remains unknown. 59.11. REFERENCES 215

[9] The equivalence between the two expressions in this equation, one in terms of the median operation and the other in terms of lattice operations and inequalities is Theorem 1 of Birkhoﬀ & Kiss (1947).

[10] Birkhoﬀ & Kiss (1947), Theorem 2.

[11] Birkhoﬀ & Kiss (1947), p. 751.

[12] Avann (1961).

[13] Knuth (2008) calls such a set an ideal, but a convex set in the graph of a distributive lattice is not the same thing as an ideal of the lattice.

[14] Imrich & Klavžar (2000), Theorem 2.40, p. 77.

[15] Bandelt & Chepoi (2008), Proposition 2.5, p.8; Chung, Graham & Saks (1989); Feder (1995); Knuth (2008), Theorem S, p. 72.

[16] Hell (1976).

[17] Imrich & Klavžar (2000), Proposition 1.33, p. 27.

[18] Bandelt (1984); Imrich & Klavžar (2000), Theorem 2.39, p.76; Knuth (2008), p. 74.

[19] The technique, which culminates in Lemma 7.10 on p.218 of Imrich and Klavžar, consists of applying an algorithm of Chiba & Nishizeki (1985) to list all 4-cycles in the graph G, forming an undirected graph having as its vertices the edges of G and having as its edges the opposite sides of a 4-cycle, and using the connected components of this derived graph to form hypercube coordinates. An equivalent algorithm is Knuth (2008), Algorithm H, p. 69.

[20] For previous median graph recognition algorithms, see Jha & Slutzki (1992), Imrich & Klavžar (1998), and Hagauer, Imrich & Klavžar (1999). For triangle detection algorithms, see Itai & Rodeh (1978), Chiba & Nishizeki (1985), and Alon, Yuster & Zwick (1995).

[21] Alon, Yuster & Zwick (1995), based on fast matrix multiplication. Here m is the number of edges in the graph, and the big O notation hides a large constant factor; the best practical algorithms for triangle detection take time O(m3/2). For median graph recognition, the time bound can be expressed either in terms of m or n (the number of vertices), as m = O(n log n).

[22] Mulder & Schrijver (1979) described a version of this method for systems of characteristics not requiring any latent vertices, and Barthélémy (1989) gives the full construction. The Buneman graph name is given in Dress et al. (1997) and Dress, Huber & Moulton (1997).

[23] Mulder & Schrijver (1979).

[24] Škrekovski (2001).

[25] Mulder (1980).

59.11 References

• Alon, Noga; Yuster, Raphael; Zwick, Uri (1995), “Color-coding”, Journal of the Association for Computing Machinery 42 (4): 844–856, doi:10.1145/210332.210337, MR 1411787. • Avann, S. P. (1961), “Metric ternary distributive semi-lattices”, Proceedings of the American Mathematical Society (American Mathematical Society) 12 (3): 407–414, doi:10.2307/2034206, JSTOR 2034206, MR 0125807. • Bandelt, Hans-Jürgen (1984), “Retracts of hypercubes”, Journal of Graph Theory 8 (4): 501–510, doi:10.1002/jgt.3190080407, MR 0766499. • Bandelt, Hans-Jürgen; Barthélémy, Jean-Pierre (1984), “Medians in median graphs”, Discrete Applied Mathe- matics 8 (2): 131–142, doi:10.1016/0166-218X(84)90096-9, MR 0743019. • Bandelt, Hans-Jürgen; Chepoi, V. (2008), “Metric graph theory and geometry: a survey” (PDF), Contemporary Mathematics, to appear. • Bandelt, Hans-Jürgen; Forster, P.; Sykes, B. C.; Richards, Martin B. (October 1, 1995), “Mitochondrial por- traits of human populations using median networks”, Genetics 141 (2): 743–753, PMC 1206770, PMID 8647407. 216 CHAPTER 59. MEDIAN GRAPH

• Bandelt, Hans-Jürgen; Forster, P.; Rohl, Arne (January 1, 1999), “Median-joining networks for inferring in- traspecific phylogenies”, Molecular Biology and Evolution 16 (1): 37–48, doi:10.1093/oxfordjournals.molbev.a026036, PMID 10331250. • Bandelt, Hans-Jürgen; Macaulay, Vincent; Richards, Martin B. (2000), “Median networks: speedy construction and greedy reduction, one simulation, and two case studies from human mtDNA”, Molecular Phylogenetics and Evolution 16 (1): 8–28, doi:10.1006/mpev.2000.0792, PMID 10877936. • Barthélémy, Jean-Pierre (1989), “From copair hypergraphs to median graphs with latent vertices”, Discrete Mathematics 76 (1): 9–28, doi:10.1016/0012-365X(89)90283-5, MR 1002234. • Birkhoff, Garrett; Kiss, S. A. (1947), “A ternary operation in distributive lattices”, Bulletin of the American Mathematical Society 53 (1): 749–752, doi:10.1090/S0002-9904-1947-08864-9, MR 0021540. • Buneman, P. (1971), “The recovery of trees from measures of dissimilarity”, in Hodson, F. R.; Kendall, D. G.; Tautu, P. T., Mathematics in the Archaeological and Historical Sciences, Edinburgh University Press, pp. 387–395. • Chepoi, V.; Dragan, F.; Vaxès, Y. (2002), “Center and diameter problems in planar quadrangulations and triangulations”, Proc. 13th ACM-SIAM Symposium on Discrete Algorithms, pp. 346–355. • Chepoi, V.; Fanciullini, C.; Vaxès, Y. (2004), “Median problem in some plane triangulations and quadrangulations”, Computational Geometry: Theory & Applications 27: 193–210, doi:10.1016/j.comgeo.2003.11.002. • Chiba, N.; Nishizeki, T. (1985), “Arboricity and subgraph listing algorithms”, SIAM Journal on Computing 14: 210–223, doi:10.1137/0214017, MR 0774940. • Chung, F. R. K.; Graham, R. L.; Saks, M. E. (1987), “Dynamic search in graphs”, in Wilf, H., Discrete Algorithms and Complexity (Kyoto, 1986) (PDF), Perspectives in Computing 15, New York: Academic Press, pp. 351–387, MR 0910939. • Chung, F. R. K.; Graham, R. L.; Saks, M. E. (1989), “A dynamic location problem for graphs” (PDF), Com- binatorica 9 (2): 111–132, doi:10.1007/BF02124674. • Day, William H. E.; McMorris, F. R. (2003), Axiomatic Consensus Theory in Group Choice and Bioinformatics, Society for Industrial and Applied Mathematics, pp. 91–94, ISBN 0-89871-551-2. • Dress, A.; Hendy, M.; Huber, K.; Moulton, V. (1997), “On the number of vertices and edges of the Buneman graph”, Annals of Combinatorics 1 (1): 329–337, doi:10.1007/BF02558484, MR 1630739. • Dress, A.; Huber, K.; Moulton, V. (1997), “Some variations on a theme by Buneman”, Annals of Combinatorics 1 (1): 339–352, doi:10.1007/BF02558485, MR 1630743. • Duffus, Dwight; Rival, Ivan (1983), “Graphs orientable as distributive lattices”, Proceedings of the Ameri- can Mathematical Society (American Mathematical Society) 88 (2): 197–200, doi:10.2307/2044697, JSTOR 2044697. • Feder, T. (1995), Stable Networks and Product Graphs, Memoirs of the American Mathematical Society 555. • Hagauer, Johann; Imrich, Wilfried; Klavžar, Sandi (1999), “Recognizing median graphs in subquadratic time”, Theoretical Computer Science 215 (1–2): 123–136, doi:10.1016/S0304-3975(97)00136-9, MR 1678773. • Hell, Pavol (1976), “Graph retractions”, Colloquio Internazionale sulle Teorie Combinatorie (Roma, 1973), Tomo II, Atti dei Convegni Lincei 17, Rome: Accad. Naz. Lincei, pp. 263–268, MR 0543779. • Imrich, Wilfried; Klavžar, Sandi (1998), “A convexity lemma and expansion procedures for bipartite graphs”, European Journal of Combinatorics 19 (6): 677–686, doi:10.1006/eujc.1998.0229, MR 1642702. • Imrich, Wilfried; Klavžar, Sandi (2000), Product Graphs: Structure and Recognition, Wiley, ISBN 0-471- 37039-8, MR 788124. • Imrich, Wilfried; Klavžar, Sandi; Mulder, Henry Martyn (1999), “Median graphs and triangle-free graphs”, SIAM Journal on Discrete Mathematics 12 (1): 111–118, doi:10.1137/S0895480197323494, MR 1666073. • Itai, A.; Rodeh, M. (1978), “Finding a minimum circuit in a graph”, SIAM Journal on Computing 7 (4): 413– 423, doi:10.1137/0207033, MR 0508603. 59.12. EXTERNAL LINKS 217

• Jha, Pranava K.; Slutzki, Giora (1992), “Convex-expansion algorithms for recognizing and isometric embedding of median graphs”, Ars Combinatoria 34: 75–92, MR 1206551. • Klavžar, Sandi; Mulder, Henry Martyn (1999), “Median graphs: characterizations, location theory and related structures”, Journal of Combinatorial Mathematics and Combinatorial Computing 30: 103–127, MR 1705337. • Klavžar, Sandi; Mulder, Henry Martyn; Škrekovski, Riste (1998), “An Euler-type formula for median graphs”, Discrete Mathematics 187 (1): 255–258, doi:10.1016/S0012-365X(98)00019-3, MR 1630736. • Knuth, Donald E. (2008), “Median algebras and median graphs”, The Art of Computer Programming, IV, Fascicle 0: Introduction to Combinatorial Algorithms and Boolean Functions, Addison-Wesley, pp. 64–74, ISBN 978-0-321-53496-5.

• Mulder, Henry Martyn (1980), "n-cubes and median graphs”, Journal of Graph Theory 4 (1): 107–110, doi:10.1002/jgt.3190040112, MR 0558458.

• Mulder, Henry Martyn; Schrijver, Alexander (1979), “Median graphs and Helly hypergraphs”, Discrete Math- ematics 25 (1): 41–50, doi:10.1016/0012-365X(79)90151-1, MR 0522746. • Nebeský, Ladislav (1971), “Median graphs”, Commentationes Mathematicae Universitatis Carolinae 12: 317– 325, MR 0286705. • Škrekovski, Riste (2001), “Two relations for median graphs”, Discrete Mathematics 226 (1): 351–353, doi:10.1016/S0012- 365X(00)00120-5, MR 1802603. • Soltan, P.; Zambitskii, D.; Prisăcaru, C. (1973), Extremal problems on graphs and algorithms of their solution (in Russian), Chişinău: Ştiinţa.

59.12 External links

• Median graphs, Information System for Graph Class Inclusions.

• Network, Free Phylogenetic Network Software. Network generates evolutionary trees and networks from genetic, linguistic, and other data.

• PhyloMurka, open-source software for median network computations from biological data. 218 CHAPTER 59. MEDIAN GRAPH

The graph of a distributive lattice, drawn as a Hasse diagram. 59.12. EXTERNAL LINKS 219

Retraction of a cube onto a six-vertex subgraph.

Converting a triangle-free graph into a median graph. 220 CHAPTER 59. MEDIAN GRAPH

tailless tailed

small large

brown

latent: silver small tailed silver

The Buneman graph for ﬁve types of mouse.

The Cartesian product of graphs forms a median graph from two smaller median graphs. Chapter 60

Moore graph

In graph theory, a Moore graph is a regular graph of degree d and diameter k whose number of vertices equals the upper bound

k∑−1 1 + d (d − 1)i. i=0

An equivalent definition of a Moore graph is that it is a graph of diameter k with girth 2k + 1. Another equivalent n − definition of a Moore graph G is that it has girth g = 2k+1 and precisely g (m n + 1) cycles of length g, where n,m is the number of vertices (resp. edges) of G. They are in fact extremal with respect to the number of cycles whose length is the girth of the graph (Azarija & Klavžar 2015). Moore graphs were named by Hoffman & Singleton (1960) after Edward F. Moore, who posed the question of describing and classifying these graphs. As well as having the maximum possible number of vertices for a given combination of degree and diameter, Moore graphs have the minimum possible number of vertices for a regular graph with given degree and girth. That is, any Moore graph is a cage (Erdős, Rényi & Sós 1966). The formula for the number of vertices in a Moore graph can be generalized to allow a definition of Moore graphs with even girth as well as odd girth, and again these graphs are cages.

60.1 Bounding vertices by degree and diameter

Let G be any graph with maximum degree d and diameter k, and consider the tree formed by breadth ﬁrst search starting from any vertex v. This tree has 1 vertex at level 0 (v itself), and at most d vertices at level 1 (the neighbors of v). In the next level, there are at most d(d−1) vertices: each neighbor of v uses one of its adjacencies to connect to v and so can have at most d−1 neighbors at level 2. In general, a similar argument shows that at any level 1 ≤ i ≤ k, there can be at most d(d−1)i vertices. Thus, the total number of vertices can be at most

k∑−1 1 + d (d − 1)i. i=0

Hoffman & Singleton (1960) originally defined a Moore graph as a graph for which this bound on the number of vertices is met exactly. Therefore, any Moore graph has the maximum number of vertices possible among all graphs with maximum degree d and diameter k. Later, Singleton (1968) showed that Moore graphs can equivalently be defined as having diameter k and girth 2k+1; these two requirements combine to force the graph to be d-regular for some d and to satisfy the vertex-counting formula.

221 222 CHAPTER 60. MOORE GRAPH

The Petersen graph as a Moore graph. Any breadth ﬁrst search tree has d(d−1)i vertices in its ith level.

60.2 Moore graphs as cages

Instead of upper bounding the number of vertices in a graph in terms of its maximum degree and its diameter, we can calculate via similar methods a lower bound on the number of vertices in terms of its minimum degree and its girth (Erdős, Rényi & Sós 1966). Suppose G has minimum degree d and girth 2k+1. Choose arbitrarily a starting vertex v, and as before consider the breadth-ﬁrst search tree rooted at v. This tree must have one vertex at level 0 (v itself), and at least d vertices at level 1. At level 2 (for k > 1), there must be at least d(d−1) vertices, because each vertex at level 1 has at least d−1 remaining adjacencies to ﬁll, and no two vertices at level 1 can be adjacent to each other or to a shared vertex at level 2 because that would create a cycle shorter than the assumed girth. In general, a similar argument shows that at any level 1 ≤ i ≤ k, there must be at least d(d−1)i vertices. Thus, the total number of vertices must be at least

k∑−1 1 + d (d − 1)i. i=0

In a Moore graph, this bound on the number of vertices is met exactly. Each Moore graph has girth exactly 2k+1: it does not have enough vertices to have higher girth, and a shorter cycle would cause there to be too few vertices in the first k levels of some breadth first search tree. Therefore, any Moore graph has the minimum number of vertices possible among all graphs with minimum degree d and diameter k: it is a cage. For even girth 2k, one can similarly form a breadth-first search tree starting from the midpoint of a single edge. The resulting bound on the minimum number of vertices in a graph of this girth with minimum degree d is 60.3. EXAMPLES 223

k∑−1 k∑−2 2 (d − 1)i = 1 + (d − 1)k−1 + d (d − 1)i. i=0 i=0 (The right hand side of the formula instead counts the number of vertices in a breadth ﬁrst search tree starting from a single vertex, accounting for the possibility that a vertex in the last level of the tree may be adjacent to d vertices in the previous level.) Thus, the Moore graphs are sometimes deﬁned as including the graphs that exactly meet this bound. Again, any such graph must be a cage.

60.3 Examples

The Hoﬀman–Singleton theorem states that any Moore graph with girth 5 must have degree 2, 3, 7, or 57. The Moore graphs are:

• The complete graphs Kn on n > 2 nodes. (diameter 1, girth 3, degree n-1, order n)

• The odd cycles C2n+1 . (diameter n, girth 2n+1, degree 2, order 2n+1) • The Petersen graph. (diameter 2, girth 5, degree 3, order 10) • The Hoﬀman–Singleton graph. (diameter 2, girth 5, degree 7, order 50) • A hypothetical graph of diameter 2, girth 5, degree 57 and order 3250; it is currently unknown whether such a graph exists.

Unlike all other Moore graphs, Higman proved that the unknown Moore graph cannot be vertex-transitive. Mačaj and Širáň further proved that the order of the automorphism group of such a graph is at most 375. If the generalized deﬁnition of Moore graphs that allows even girth graphs is used, the even girth Moore graphs correspond to incidence graphs of (possible degenerate) Generalized polygons. Some examples are the even cycles C2n , the complete bipartite graphs Kn,n with girth four, the Heawood graph with degree 3 and girth 6, and the Tutte–Coxeter graph with degree 3 and girth 8. More generally, it is known (Bannai & Ito 1973; Damerell 1973) that, other than the graphs listed above, all Moore graphs must have girth 5, 6, 8, or 12. The even girth case also follows from the Feit-Higman theorem about possible values of n for a generalized n-gon.

60.4 See also

• Degree diameter problem • Table of the largest known graphs of a given diameter and maximal degree

60.5 References

• Azarija, Jernej; Klavžar, Sandi (2015), “Moore Graphs and Cycles Are Extremal Graphs for Convex Cycles”, Journal of Graph Theory 80: 34–42, doi:10.1002/jgt.21837 • Bollobás, Béla (1998), “Chap.VIII.3”, Modern graph theory, Graduate Texts in Mathematics 184, Springer- Verlag. • Bannai, E.; Ito, T. (1973), “On ﬁnite Moore graphs”, Journal of the Faculty of Science, the University of Tokyo. Sect. 1 A, Mathematics 20: 191–208, MR 0323615 • Damerell, R. M. (1973), “On Moore graphs”, Mathematical Proceedings of the Cambridge Philosophical Society 74: 227–236, doi:10.1017/S0305004100048015, MR 0318004 • Erdős, Paul; Rényi, Alfréd; Sós, Vera T. (1966), “On a problem of graph theory” (PDF), Studia Sci. Math. Hungar. 1: 215–235. 224 CHAPTER 60. MOORE GRAPH

• Hoﬀman, Alan J.; Singleton, Robert R. (1960), “Moore graphs with diameter 2 and 3”, IBM Journal of Research and Development 5 (4): 497–504, doi:10.1147/rd.45.0497, MR 0140437 • Mačaj, Martin; Širáň, Jozef (2010), “Search for properties of the missing Moore graph”, Linear Algebra and its Applications 432: 2381–2398. • Singleton, Robert R. (1968), “There is no irregular Moore graph”, American Mathematical Monthly 75 (1): 42–43, doi:10.2307/2315106, MR 0225679

60.6 External links

• Brouwer and Haemers: Spectra of graphs • Weisstein, Eric W., “Moore Graph”, MathWorld.

• Weisstein, Eric W., “Hoﬀman-Singleton Theorem”, MathWorld. Chapter 61

Multipartite graph

In graph theory, a part of mathematics, a k-partite graph is a graph whose vertices are or can be partitioned into k different independent sets. Equivalently, it is a graph that can be colored with k colors, so that no two endpoints of an edge have the same color. When k = 2 these are the bipartite graphs, and when k = 3 they are called the tripartite graphs. Bipartite graphs may be recognized in polynomial time but, for any k > 2 it is NP-complete, given an uncolored graph, to test whether it is k-partite.[1] However, in some applications of graph theory, a k-partite graph may be given as input to a computation with its coloring already determined; this can happen when the sets of vertices in the graph represent different types of objects. For instance, folksonomies have been modeled mathematically by tripartite graphs in which the three sets of vertices in the graph represent users of a system, resources that the users are tagging, and tags that the users have applied to the resources.[2] A complete k-partite graph is a k-partite graph in which there is an edge between every pair of vertices from different independent sets. These graphs are described by notation with a capital letter K subscripted by a sequence of the sizes of each set in the partition. For instance, K₂,₂,₂ is the complete tripartite graph of a regular octahedron, which can be partitioned into three independent sets each consisting of two opposite vertices. A complete multipartite graph is a graph that is complete k-partite for some k.[3] The Turán graphs are the special case of complete multipartite graphs in which each two independent sets differ in size by at most one vertex. Complete k-partite graphs and complete multipartite graphs are special cases of cographs, and can be recognized in polynomial time even when the partition is not supplied as part of the input.

61.1 References

[1] Garey, M. R.; Johnson, D. S. (1979), Computers and Intractability: A Guide to the Theory of NP-Completeness, W.H. Freeman, GT4, ISBN 0-7167-1045-5.

[2] Hotho, Andreas; Jäschke, Robert; Schmitz, Christoph; Stumme, Gerd (2006), “FolkRank : A Ranking Algorithm for Folksonomies”, LWA 2006: Lernen - Wissensentdeckung - Adaptivität, Hildesheim, October 9th-11th 2006, pp. 111–114.

[3] Chartrand, Gary; Zhang, Ping (2008), Chromatic Graph Theory, CRC Press, p. 41, ISBN 9781584888017.

225 226 CHAPTER 61. MULTIPARTITE GRAPH

The complete tripartite graph K2,2,2, the graph of the octahedron Chapter 62

Null graph

In the mathematical ﬁeld of graph theory, the term “null graph” may refer either to the order-zero graph, or alternatively, to any edgeless graph (the latter is sometimes called an “empty graph”).

62.1 Order-zero graph

The order-zero graph, K0 , is the unique graph having no vertices (hence its order is zero). It follows that K0 also has no edges. Some authors exclude K0 from consideration as a graph (either by definition, or more simply as a matter of convenience). Whether including K0 as a valid graph is useful depends on context. On the positive side, K0 follows naturally from the usual set-theoretic definitions of a graph (it is the ordered pair (V, E) for which the vertex and edge sets, V and E, are both empty), in proofs it serves as a natural base case for mathematical induction, and similarly, in recursively defined data structures K0 is useful for defining the base case for recursion (by treating the null tree as the child of missing edges in any non-null binary tree, every non-null binary tree has exactly two children). On the negative side, including K0 as a graph requires that many well-defined formulas for graph properties include exceptions for it (for example, either “counting all strongly connected components of a graph” becomes “counting all non-null strongly connected components of a graph”, or the definition of connected graphs has to be modified not to include K0). To avoid the need for such exceptions, it is often assumed in literature that the term graph implies “graph with at least one vertex” unless context suggests otherwise.[1][2] In category theory, the order-zero graph is, according to some definitions of “category of graphs,” the initial object in the category.

K0 does fulﬁll (vacuously) most of the same basic graph properties as does K1 (the graph with one vertex and no edges). As some examples, K0 is of size zero, it is equal to its complement graph K0 , a forest, and a planar graph. It may be considered undirected, directed, or even both; when considered as directed, it is a directed acyclic graph. And it is both a complete graph and an edgeless graph. However, deﬁnitions for each of these graph properties will vary depending on whether context allows for K0 .

62.2 Edgeless graph

For each natural number n, the edgeless graph (or empty graph) Kn of order n is the graph with n vertices and zero edges. An edgeless graph is occasionally referred to as a null graph in contexts where the order-zero graph is not permitted.[1][2]

The notation Kn arises from the fact that the n-vertex edgeless graph is the complement of the complete graph Kn .

62.3 See also

• Glossary of graph theory

• Cycle graph

227 228 CHAPTER 62. NULL GRAPH

• Path graph

62.4 Notes

[1] Weisstein, Eric W., “Empty Graph”, MathWorld.

[2] Weisstein, Eric W., “Null Graph”, MathWorld.

62.5 References

• Harary, F. and Read, R. (1973), “Is the null graph a pointless concept?", Graphs and Combinatorics (Confer- ence, George Washington University), Springer-Verlag, New York, NY. Chapter 63

Outerplanar graph

A maximal outerplanar graph and its 3-coloring.

In graph theory, an outerplanar graph is a graph that has a planar drawing for which all vertices belong to the outer

229 230 CHAPTER 63. OUTERPLANAR GRAPH

The complete graph K4 is the smallest planar graph that is not outerplanar.

face of the drawing. Outerplanar graphs may be characterized (analogously to Wagner’s theorem for planar graphs) by the two forbidden minors K4 and K₂,₃, or by their Colin de Verdière graph invariants. They have Hamiltonian cycles if and only if they are biconnected, in which case the outer face forms the unique Hamiltonian cycle. Every outerplanar graph is 3-colorable, and has degeneracy and treewidth at most 2. The outerplanar graphs are a subset of the planar graphs, the subgraphs of series-parallel graphs, and the circle graphs. The maximal outerplanar graphs, those two which no more edges can be added while preserving outerplanarity, are also chordal graphs and visibility graphs.

63.1 History

Outerplanar graphs were ﬁrst studied and named by Chartrand & Harary (1967), in connection with the problem of determining the planarity of graphs formed by using a perfect matching to connect two copies of a base graph (for instance, many of the generalized Petersen graphs are formed in this way from two copies of a cycle graph). As they showed, when the base graph is biconnected, a graph constructed in this way is planar if and only if its base graph is outerplanar and the matching forms a dihedral permutation of its outer cycle. 63.2. DEFINITION AND CHARACTERIZATIONS 231

63.2 Deﬁnition and characterizations

An outerplanar graph is an undirected graph that can be drawn in the plane without crossings in such a way that all of the vertices belong to the unbounded face of the drawing. That is, no vertex is totally surrounded by edges. Alternatively, a graph G is outerplanar if the graph formed from G by adding a new vertex, with edges connecting it to all the other vertices, is a planar graph.[1] A maximal outerplanar graph is an outerplanar graph that cannot have any additional edges added to it while preserving outerplanarity. Every maximal outerplanar graph with n vertices has exactly 2n − 3 edges, and every bounded face of a maximal outerplanar graph is a triangle.

63.2.1 Forbidden graphs

Outerplanar graphs have a forbidden graph characterization analogous to Kuratowski’s theorem and Wagner’s theorem for planar graphs: a graph is outerplanar if and only if it does not contain a subdivision of the complete graph K4 or [2] the complete bipartite graph K₂,₃. Alternatively, a graph is outerplanar if and only if it does not contain K4 or K₂,₃ as a minor, a graph obtained from it by deleting and contracting edges.[3] A triangle-free graph is outerplanar if and only if it does not contain a subdivision of K₂,₃.[4]

63.2.2 Colin de Verdière invariant

A graph is outerplanar if and only if its Colin de Verdière graph invariant is at most two. The graphs characterized in a similar way by having Colin de Verdière invariant at most one, three, or four are respectively the linear forests, planar graphs, and linklessly embeddable graphs.

63.3 Properties

63.3.1 Biconnectivity and Hamiltonicity

An outerplanar graph is biconnected if and only if the outer face of the graph forms a simple cycle without repeated vertices. An outerplanar graph is Hamiltonian if and only if it is biconnected; in this case, the outer face forms the unique Hamiltonian cycle.[5] More generally, the size of the longest cycle in an outerplanar graph is the same as the number of vertices in its largest biconnected component. For this reason ﬁnding Hamiltonian cycles and longest cycles in outerplanar graphs may be solved in linear time, in contrast to the NP-completeness of these problems for arbitrary graphs. Every maximal outerplanar graph satisﬁes a stronger condition than Hamiltonicity: it is node pancyclic, meaning that for every vertex v and every k in the range from three to the number of vertices in the graph, there is a length-k cycle containing v. A cycle of this length may be found by repeatedly removing a triangle that is connected to the rest of the graph by a single edge, such that the removed vertex is not v, until the outer face of the remaining graph has length k.[6] A planar graph is outerplanar if and only if each of its biconnected components is outerplanar.[4]

63.3.2 Coloring

All loopless outerplanar graphs can be colored using only three colors;[7] this fact features prominently in the simplified proof of Chvátal’s art gallery theorem by Fisk (1978). A 3-coloring may be found in linear time by a greedy coloring algorithm that removes any vertex of degree at most two, colors the remaining graph recursively, and then adds back the removed vertex with a color different from the colors of its two neighbors. According to Vizing’s theorem, the chromatic index of any graph (the minimum number of colors needed to color its edges so that no two adjacent edges have the same color) is either the maximum degree of any vertex of the graph or one plus the maximum degree. However, in an outerplanar graph, the chromatic index is equal to the maximum degree except when the graph forms a cycle of odd length.[8] An edge coloring with an optimal number of colors can be found in linear time based on a breadth-first traversal of the weak dual tree.[7] 232 CHAPTER 63. OUTERPLANAR GRAPH

63.3.3 Other properties

Outerplanar graphs have degeneracy at most two: every subgraph of an outerplanar graph contains a vertex with degree at most two.[9] Outerplanar graphs have treewidth at most two, which implies that many graph optimization problems that are NP- complete for arbitrary graphs may be solved in polynomial time by dynamic programming when the input is outerplanar. More generally, k-outerplanar graphs have treewidth O(k).[10] Every outerplanar graph can be represented as an intersection graph of axis-aligned rectangles in the plane, so outerplanar graphs have boxicity at most two.[11]

63.4 Related families of graphs

Every outerplanar graph is a planar graph. Every outerplanar graph is also a subgraph of a series-parallel graph.[12] However, not all planar series-parallel graphs are outerplanar. The complete bipartite graph K₂,₃ is planar and series- parallel but not outerplanar. On the other hand, the complete graph K4 is planar but neither series-parallel nor outerplanar. Every forest and every cactus graph are outerplanar.[13] The weak planar dual graph of an embedded outerplanar graph (the graph that has a vertex for every bounded face of the embedding, and an edge for every pair of adjacent bounded faces) is a forest, and the weak planar dual of a Halin graph is an outerplanar graph. A planar graph is outerplanar if and only if its weak dual is a forest, and it is Halin if and only if its weak dual is biconnected and outerplanar.[14] There is a notion of degree of outerplanarity. A 1-outerplanar embedding of a graph is the same as an outerplanar embedding. For k > 1 a planar embedding is said to be k-outerplanar if removing the vertices on the outer face results in a (k − 1)-outerplanar embedding. A graph is k-outerplanar if it has a k-outerplanar embedding.[15] Every maximal outerplanar graph is a chordal graph. Every maximal outerplanar graph is the visibility graph of a simple polygon.[16] Maximal outerplanar graphs are also formed as the graphs of polygon triangulations. They are examples of 2-trees, of series-parallel graphs, and of chordal graphs. Every outerplanar graph is a circle graph, the intersection graph of a set of chords of a circle.[17]

63.5 Notes

[1] Felsner (2004).

[2] Chartrand & Harary (1967); Sysło (1979); Brandstädt, Le & Spinrad (1999), Proposition 7.3.1, p. 117; Felsner (2004).

[3] Diestel (2000).

[4] Sysło (1979).

[5] Chartrand & Harary (1967); Sysło (1979).

[6] Li, Corneil & Mendelsohn (2000), Proposition 2.5.

[7] Proskurowski & Sysło (1986).

[8] Fiorini (1975).

[9] Lick & White (1970).

[10] Baker (1994).

[11] Scheinerman (1984); Brandstädt, Le & Spinrad (1999), p. 54.

[12] Brandstädt, Le & Spinrad (1999), p. 174.

[13] Brandstädt, Le & Spinrad (1999), p. 169.

[14] Sysło & Proskurowski (1983).

[15] Kane & Basu (1976); Sysło (1979). 63.5. NOTES 233

A cactus graph. The cacti form a subclass of the outerplanar graphs. 234 CHAPTER 63. OUTERPLANAR GRAPH

[16] El-Gindy (1985); Lin & Skiena (1995); Brandstädt, Le & Spinrad (1999), Theorem 4.10.3, p. 65.

[17] Wessel & Pöschel (1985); Unger (1988).

63.6 References

• Baker, Brenda S. (1994), “Approximation algorithms for NP-complete problems on planar graphs”, Journal of the ACM 41 (1): 153–180, doi:10.1145/174644.174650.

• Boza, Luis; Fedriani, Eugenio M.; Núñez, Juan (2004), “The problem of outer embeddings in pseudosurfaces”, Ars Combinatoria 71: 79–91.

• Boza, Luis; Fedriani, Eugenio M.; Núñez, Juan (2004), “Obstruction sets for outer-bananas-surface graphs”, Ars Combinatoria 73: 65–77.

• Boza, Luis; Fedriani, Eugenio M.; Núñez, Juan (2006), “Uncountable graphs with all their vertices in one face”, Acta Mathematica Hungarica, 112 (4): 307–313, doi:10.1007/s10474-006-0082-0.

• Boza, Luis; Fedriani, Eugenio M.; Núñez, Juan (2010), “Outer-embeddability in certain pseudosurfaces arising from three spheres”, Discrete Mathematics 310: 3359–3367, doi:10.1016/j.disc.2010.07.027.

• Brandstädt, Andreas; Le, Van Bang; Spinrad, Jeremy (1999), Graph Classes: A Survey, SIAM Monographs on Discrete Mathematics and Applications, Society for Industrial and Applied Mathematics, ISBN 0-89871-432- X.

• Chartrand, Gary; Harary, Frank (1967), “Planar permutation graphs”, Annales de l'Institut Henri Poincaré B 3 (4): 433–438.

• Diestel, Reinhard (2000), Graph Theory, Graduate Texts in Mathematics 173, Springer-Verlag, p. 107, ISBN 0-387-98976-5.

• El-Gindy, H. (1985), Hierarchical decomposition of polygons with applications, Ph.D. thesis, McGill University. As cited by Brandstädt, Le & Spinrad (1999).

• Felsner, Stefan (2004), Geometric graphs and arrangements: some chapters from combinational geometry, Vieweg+Teubner Verlag, p. 6, ISBN 978-3-528-06972-8.

• Fiorini, Stanley (1975), “On the chromatic index of outerplanar graphs”, Journal of Combinatorial Theory, Series B 18 (1): 35–38, doi:10.1016/0095-8956(75)90060-X.

• Fisk, Steve (1978), “A short proof of Chvátal’s watchman theorem”, Journal of Combinatorial Theory, Series B 24: 374, doi:10.1016/0095-8956(78)90059-X.

• Fleischner, Herbert J.; Geller, D. P.; Harary, Frank (1974), “Outerplanar graphs and weak duals”, Journal of the Indian Mathematical Society 38: 215–219, MR 0389672.

• Kane, Vinay G.; Basu, Sanat K. (1976), “On the depth of a planar graph”, Discrete Mathematics 14 (1): 63–67, doi:10.1016/0012-365X(76)90006-6.

• Li, Ming-Chu; Corneil, Derek G.; Mendelsohn, Eric (2000), “Pancyclicity and NP-completeness in planar graphs”, Discrete Applied Mathematics 98 (3): 219–225, doi:10.1016/S0166-218X(99)00163-8.

• Lick, Don R.; White, Arthur T. (1970), “k-degenerate graphs”, Canadian Journal of Mathematics 22: 1082– 1096, doi:10.4153/CJM-1970-125-1.

• Lin, Yaw-Ling; Skiena, Steven S. (1995), “Complexity aspects of visibility graphs”, International Journal of Computational Geometry and Applications 5 (3): 289–312, doi:10.1142/S0218195995000179.

• Proskurowski, Andrzej; Sysło, Maciej M. (1986), “Eﬃcient vertex-and edge-coloring of outerplanar graphs”, SIAM Journal on Algebraic and Discrete Methods 7: 131–136, doi:10.1137/0607016.

• Scheinerman, E. R. (1984), Intersection Classes and Multiple Intersection Parameters of a Graph, Ph.D. thesis, Princeton University. As cited by Brandstädt, Le & Spinrad (1999). 63.7. EXTERNAL LINKS 235

• Sysło, Maciej M. (1979), “Characterizations of outerplanar graphs”, Discrete Mathematics 26 (1): 47–53, doi:10.1016/0012-365X(79)90060-8. • Sysło, Maciej M.; Proskurowski, Andrzej (1983), “On Halin graphs”, Graph Theory: Proceedings of a Con- ference held in Lagów, Poland, February 10–13, 1981, Lecture Notes in Mathematics 1018, Springer-Verlag, pp. 248–256, doi:10.1007/BFb0071635.

• Unger, Walter (1988), “On the k-colouring of circle-graphs”, Proc. 5th Symposium on Theoretical Aspects of Computer Science (STACS '88), Lecture Notes in Computer Science 294, Springer-Verlag, pp. 61–72, doi:10.1007/BFb0035832. • Wessel, W.; Pöschel, R. (1985), “On circle graphs”, in Sachs, Horst, Graphs, Hypergraphs and Applications: Proceedings of the Conference on Graph Theory Held in Eyba, October 1st to 5th, 1984, Teubner-Texte zur Mathematik 73, B.G. Teubner, pp. 207–210. As cited by Unger (1988).

63.7 External links

• Outerplanar graphs, Information System on Graph Classes and Their Inclusions

• Weisstein, Eric W., “Outplanar Graph”, MathWorld. Chapter 64

Overfull graph

In graph theory, an overfull graph is a graph whose size is greater than the product of its maximum degree and half of its order ﬂoored, i.e. |E| > ∆(G)⌊|V |/2⌋ where |E| is the size of G, ∆(G) is the maximum degree of G, and |V | is the order of G. The concept of an overfull subgraph, an overfull graph that is a subgraph, immediately follows. An alternate, stricter deﬁnition of an overfull subgraph S of a graph G requires ∆(G) = ∆(S) .

64.1 Properties

A few properties of overfull graphs:

1. Overfull graphs are of odd order.

2. Overfull graphs are class 2. That is, they require at least Δ + 1 colors in any edge coloring.

3. A graph G, with an overfull subgraph S such that ∆(G) = ∆(S) , is of class 2.

64.2 Overfull conjecture

In 1986, Chetwynd and Hilton posited the following conjecture that is now known as the overfull conjecture.[1]

A graph G with ∆(G) ≥ n/3 is class 2 if and only if it has an overfull subgraph S such that ∆(G) = ∆(S) .

This conjecture, if true, would have numerous implications in graph theory, including the 1-factorization conjecture.[2]

64.3 Algorithms

≥ n For graphs in which ∆ 3 , there are at most three induced overfull subgraphs, and it is possible to ﬁnd an overfull ≥ n subgraph in polynomial time. When ∆ 2 , there is at most one induced overfull subgraph, and it is possible to ﬁnd it in linear time.[3]

64.4 References

[1] Chetwynd, A. G.; Hilton, A. J. W. (1986), “Star multigraphs with three vertices of maximum degree”, Mathematical Proceedings of the Cambridge Philosophical Society 100 (2): 303–317, doi:10.1017/S030500410006610X, MR 848854.

[2] Chetwynd, A. G.; Hilton, A. J. W. (1989), “1-factorizing regular graphs of high degree—an improved bound”, Discrete Mathematics 75 (1-3): 103–112, doi:10.1016/0012-365X(89)90082-4, MR 1001390.

236 64.4. REFERENCES 237

[3] Niessen, Thomas (2001), “How to ﬁnd overfull subgraphs in graphs with large maximum degree. II”, Electronic Journal of Combinatorics 8 (1), Research Paper 7, MR 1814514. Chapter 65

Panconnectivity

In graph theory, a panconnected graph is an undirected graph in which, for every two vertices s and t, there exist paths from s to t of every possible length from the distance d(s,t) up to n − 1, where n is the number of vertices in the graph. The concept of panconnectivity was introduced in 1975 by Yousef Alavi and James E. Williamson.[1] Panconnected graphs are necessarily pancyclic: if uv is an edge, then it belongs to a cycle of every possible length, and therefore the graph contains a cycle of every possible length. Panconnected graphs and are also a generalization of Hamiltonian-connected graphs (graphs that have a Hamiltonian path connecting every pair of vertices). Several classes of graphs are known to be panconnected:

• If G has a Hamiltonian cycle, then the square of G (the graph on the same vertex set that has an edge between every two vertices whose distance in G is at most two) is panconnected.[1]

• If G is any connected graph, then the cube of G (the graph on the same vertex set that has an edge between every two vertices whose distance in G is at most three) is panconnected.[1]

• If every vertex in an n-vertex graph has degree at least n/2 + 1, then the graph is panconnected.[2] • If an n-vertex graph has at least (n − 1)(n − 2)/2 + 3 edges, then the graph is panconnected.[2]

65.1 References

[1] Alavi, Yousef; Williamson, James E. (1975), “Panconnected graphs”, Studia Scientiarum Mathematicarum Hungarica 10 (1–2): 19–22, MR 0450125.

[2] Williamson, James E. (1977), “Panconnected graphs. II”, Periodica Mathematica Hungarica. Journal of the János Bolyai Mathematical Society 8 (2): 105–116, doi:10.1007/BF02018497, MR 0463037.

238 Chapter 66

Pancyclic graph

Cycles of all possible lengths in the graph of an octahedron, showing it to be pancyclic.

In the mathematical study of graph theory, a pancyclic graph is a directed graph or undirected graph that contains cycles of all possible lengths from three up to the number of vertices in the graph.[1] Pancyclic graphs are a generalization of Hamiltonian graphs, graphs which have a cycle of the maximum possible length.

239 240 CHAPTER 66. PANCYCLIC GRAPH

66.1 Deﬁnitions

An n-vertex graph G is pancyclic if, for every k in the range 3 ≤ k ≤ n, G contains a cycle of length k.[1] It is node- pancyclic or vertex-pancyclic if, for every vertex v and every k in the same range, it contains a cycle of length k that contains v.[2] Similarly, it is edge-pancyclic if, for every edge e and every k in the same range, it contains a cycle of length k that contains e.[2] A bipartite graph cannot be pancyclic, because it does not contain any odd-length cycles, but it is said to be bipancyclic if it contains cycles of all even lengths from 4 to n.[3]

66.2 Planar graphs

A maximal outerplanar graph is a graph formed by a simple polygon in the plane by triangulating its interior. Every maximal outerplanar graph is pancyclic, as can be shown by induction. The outer face of the graph is an n-vertex cycle, and removing any triangle connected to the rest of the graph by only one edge (a leaf of the tree that forms the dual graph of the triangulation) forms a maximal outerplanar graph on one fewer vertex, that by induction has cycles of all the remaining lengths. With more care in choosing which triangle to remove, the same argument shows more strongly that every maximal outerplanar graph is node-pancyclic.[4] The same holds for graphs that have a maximal outerplanar spanning subgraph, as do for instance the wheel graphs. A maximal planar graph is a planar graph in which all faces, even the outer face, are triangles. A maximal planar graph is node-pancyclic if and only if it has a Hamiltonian cycle:[5] if it is not Hamiltonian, it is certainly not pancyclic, and if it is Hamiltonian, then the interior of the Hamiltonian cycle forms a maximal outerplanar graph on the same nodes, to which the previous argument for maximal outerplanar graphs can be applied.[6] For instance, the illustration shows the pancyclicity of the graph of an octahedron, a Hamiltonian maximal planar graph with six vertices. More strongly, by the same argument, if a maximal planar graph has a cycle of length k, it has cycles of all smaller lengths.[7] Halin graphs are the planar graphs formed from a planar drawing of a tree that has no degree-two vertices, by adding a cycle to the drawing that connects all the leaves of the tree. Halin graphs are not necessarily pancyclic, but they are almost pancyclic in the sense that there is at most one missing cycle length. The length of the missing cycle is necessarily even. If none of the interior vertices of a Halin graph has degree three, then it is necessarily pancyclic.[8] Bondy (1971) observed that many classical conditions for the existence of a Hamiltonian cycle were also suﬃcient conditions for a graph to be pancyclic, and on this basis conjectured that every 4-connected planar graph is pancyclic. However, Malkevitch (1971) found a family of counterexamples.

66.3 Tournaments

A tournament is a directed graph with one directed edge between each pair of vertices. Intuitively, a tournament can be used to model a round-robin sports competition, by drawing an edge from the winner to the loser of each game in the competition. A tournament is called strongly connected or strong if and only if it cannot be partitioned into two subsets L and W of losers and winners, such that every competitor in W beats every competitor in L.[9] Every strong tournament is pancyclic[10] and node-pancyclic.[11] If a tournament is regular (each competitor has the same number of wins and losses as each other competitor) then it is also edge-pancyclic;[12] however, a strong tournament with four vertices cannot be edge-pancyclic.

66.4 Graphs with many edges

Mantel’s theorem states that any n-vertex undirected graph with at least n2/4 edges, and no multiple edges or self- loops, either contains a triangle or it is the complete bipartite graph Kn/₂,n/₂. This theorem can be strengthened: any undirected Hamiltonian graph with at least n2/4 edges is either pancyclic or Kn/₂,n/₂.[1] There exist n-vertex Hamiltonian directed graphs with n(n + 1)/2 − 3 edges that are not pancyclic, but every Hamil- tonian directed graph with at least n(n + 1)/2 − 1 edges is pancyclic. Additionally, every n-vertex strongly connected graph in which each edge has degree at least n (counting incoming and outgoing edges together) is either pancyclic or it is a complete bipartite graph.[13] 66.5. GRAPH POWERS 241

66.5 Graph powers

For any graph G, its kth power Gk is deﬁned as the graph on the same vertex set that has an edge between every two vertices whose distance in G is at most k. If G is 2-vertex-connected, then by Fleischner’s theorem its square G2 is Hamiltonian; this can be strengthened to show that it is necessarily vertex-pancyclic.[14] More strongly, whenever G2 is Hamiltonian, it is also pancyclic.[15]

66.6 Computational complexity

It is NP-complete to test whether a graph is pancyclic, even for the special case of 3-connected cubic graphs, and it is also NP-complete to test whether a graph is node-pancyclic, even for the special case of polyhedral graphs.[16] It is also NP-complete to test whether the square of a graph is Hamiltonian, and therefore whether it is pancyclic.[17]

66.7 History

Pancyclicity was ﬁrst investigated in the context of tournaments by Harary & Moser (1966), Moon (1966), and Alspach (1967). The concept of pancyclicity was named and extended to undirected graphs by Bondy (1971).

66.8 Notes

[1] Bondy (1971).

[2] Randerath et al. (2002).

[3] Schmeichel & Mitchem (1982).

[4] Li, Corneil & Mendelsohn (2000), Proposition 2.5.

[5] Helden (2007), Corollary 3.78.

[6] Bernhart & Kainen (1979).

[7] Hakimi & Schmeichel (1979).

[8] Skowrońska (1985).

[9] Harary & Moser (1966), Corollary 5b.

[10] Harary & Moser (1966), Theorem 7.

[11] Moon (1966), Theorem 1.

[12] Alspach (1967).

[13] Häggkvist & Thomassen (1976).

[14] Hobbs (1976).

[15] Fleischner (1976).

[16] Li, Corneil & Mendelsohn (2000), Theorems 2.3 and 2.4.

[17] Underground (1978). 242 CHAPTER 66. PANCYCLIC GRAPH

66.9 References

• Alspach, Brian (1967), “Cycles of each length in regular tournaments”, Canadian Mathematical Bulletin 10 (2): 283–286. • Bernhart, Frank; Kainen, Paul C. (1979), “The book thickness of a graph”, Journal of Combinatorial Theory, Series B 27 (3): 320–331, doi:10.1016/0095-8956(79)90021-2.

• Bondy, J. A. (1971), “Pancyclic graphs I”, Journal of Combinatorial Theory, Series B 11 (1): 80–84, doi:10.1016/0095- 8956(71)90016-5.

• Fleischner, H. (1976), “In the square of graphs, Hamiltonicity and pancyclicity, Hamiltonian connectedness and panconnectedness are equivalent concepts”, Monatshefte für Mathematik 82 (2): 125–149, doi:10.1007/BF01305995, MR 0427135. • Häggkvist, Roland; Thomassen, Carsten (1976), “On pancyclic digraphs”, Journal of Combinatorial Theory, Series B 20 (1): 20–40, doi:10.1016/0095-8956(76)90063-0. • Hakimi, S. L.; Schmeichel, E. F. (1979), “On the number of cycles of length k in a maximal planar graph”, Journal of Graph Theory 3: 69–86, doi:10.1002/jgt.3190030108. • Harary, Frank; Moser, Leo (1966), “The theory of round robin tournaments”, American Mathematical Monthly 73 (3): 231–246, doi:10.2307/2315334. • Helden, Guido (2007), Hamiltonicity of maximal planar graphs and planar triangulations (PDF), Dissertation, Rheinisch-Westfälischen Technischen Hochschule Aachen. • Hobbs, Arthur M. (1976), “The square of a block is vertex pancyclic”, Journal of Combinatorial Theory, Series B 20 (1): 1–4, doi:10.1016/0095-8956(76)90061-7, MR 0416980. • Li, Ming-Chu; Corneil, Derek G.; Mendelsohn, Eric (2000), “Pancyclicity and NP-completeness in planar graphs”, Discrete Applied Mathematics 98 (3): 219–225, doi:10.1016/S0166-218X(99)00163-8. • Malkevitch, Joseph (1971), “On the lengths of cycles in planar graphs”, Recent Trends in Graph Theory, Lecture Notes in Mathematics 186, Springer-Verlag, pp. 191–195, doi:10.1007/BFb0059437. • Moon, J. W. (1966), “On subtournaments of a tournament”, Canadian Mathematical Bulletin 9 (3): 297–301, doi:10.4153/CMB-1966-038-7. • Randerath, Bert; Schiermeyer, Ingo; Tewes, Meike; Volkmann, Lutz (2002), “Vertex pancyclic graphs”, Dis- crete Applied Mathematics 120 (1-3): 219–237, doi:10.1016/S0166-218X(01)00292-X.

• Schmeichel, Edward; Mitchem, John (1982), “Bipartite graphs with cycles of all even lengths”, Journal of Graph Theory 6 (4): 429–439, doi:10.1002/jgt.3190060407.

• Skowrońska, Mirosława (1985), “The pancyclicity of Halin graphs and their exterior contractions”, in Alspach, Brian R.; Godsil, Christopher D., Cycles in Graphs, Annals of Discrete Mathematics 27, Elsevier Science Publishers B.V., pp. 179–194. • Underground, Paris (1978), “On graphs with Hamiltonian squares”, Discrete Mathematics 21 (3): 323, doi:10.1016/0012- 365X(78)90164-4, MR 522906.

66.10 External links

• Weisstein, Eric W., “Pancyclic Graph”, MathWorld. Chapter 67

Partial cube

In graph theory, a partial cube is a graph that is an isometric subgraph of a hypercube.[1] In other words, a partial cube is a subgraph of a hypercube that preserves distances—the distance between any two vertices in the subgraph is the same as the distance between those vertices in the hypercube. Equivalently, a partial cube is a graph whose vertices can be labeled with bit strings of equal length in such a way that the distance between two vertices in the graph is equal to the Hamming distance between their labels. Such a labeling is called a Hamming labeling; it represents an isometric embedding of the partial cube into a hypercube.

67.1 History

Firsov (1965) was the ﬁrst to study isometric embeddings of graphs into hypercubes. The graphs that admit such embeddings were characterized by Djoković (1973) and Winkler (1984), and were later named partial cubes. A separate line of research on the same structures, in the terminology of families of sets rather than of hypercube labelings of graphs, was followed by Kuzmin & Ovchinnikov (1975) and Falmagne & Doignon (1997), among others.[2]

67.2 Examples

Every tree is a partial cube. For, suppose that a tree T has m edges, and number these edges (arbitrarily) from 0 to m − 1. Choose a root vertex r for the tree, arbitrarily, and label each vertex v with a string of m bits that has a 1 in position i whenever edge i lies on the path from r to v in T. For instance, r itself will have a label that is all zero bits, its neighbors will have labels with a single 1-bit, etc. Then the Hamming distance between any two labels is the distance between the two vertices in the tree, so this labeling shows that T is a partial cube. Every hypercube graph is itself a partial cube, which can be labeled with all the diﬀerent bitstrings of length equal to the dimension of the hypercube. More complex examples include the following:

• Consider the graph whose vertex labels consist of all possible (2n + 1)-digit bitstrings that have either n or n + 1 nonzero bits, where two vertices are adjacent whenever their labels diﬀer by a single bit. This labeling deﬁnes an embedding of these graphs into a hypercube (the graph of all bitstrings of a given length, with the same adjacency-condition) that turns out to be distance-preserving. The resulting graph is a bipartite Kneser graph; the graph formed in this way with n = 2 has 20 vertices and 30 edges, and is called the Desargues graph.

• All median graphs are partial cubes.[3] The trees and hypercube graphs are examples of median graphs. Since the median graphs include the squaregraphs, simplex graphs, and Fibonacci cubes, as well as the covering graphs of ﬁnite distributive lattices, these are all partial cubes.

• The planar dual graph of an arrangement of lines in the Euclidean plane is a partial cube. More generally, for any hyperplane arrangement in Euclidean space of any number of dimensions, the graph that has a vertex for each cell of the arrangement and an edge for each two adjacent cells is a partial cube.[4]

243 244 CHAPTER 67. PARTIAL CUBE

1101001

1000001 1100001 1111001 1111011

1010001 1110001 1110011 0010001 1111101 111111111111111111110 0110001

1110101 1110111

0110101

An example of a partial cube with a Hamming labeling on its vertices. This graph is also a median graph.

• A partial cube in which every vertex has exactly three neighbors is known as a cubic partial cube. Although several inﬁnite families of cubic partial cubes are known, together with many other sporadic examples, the only known cubic partial cube that is not a planar graph is the Desargues graph.[5]

• The underlying graph of any antimatroid, having a vertex for each set in the antimatroid and an edge for every two sets that diﬀer by a single element, is always a partial cube.

• The Cartesian product of any ﬁnite set of partial cubes is another partial cube.[6]

67.3 The Djoković–Winkler relation

Many of the theorems about partial cubes are based directly or indirectly upon a certain binary relation defined on the edges of the graph. This relation, first described by Djoković (1973) and given an equivalent definition in terms of distances by Winkler (1984), is denoted by Θ . Two edges e = {x, y} and f = {u, v} are defined to be in the relation Θ , written eΘf , if d(x, u) + d(y, v) ≠ d(x, v) + d(y, u) . This relation is reflexive and symmetric, but in general it is not transitive. Winkler showed that a connected graph is a partial cube if and only if it is bipartite and the relation Θ is transitive.[7] In this case, it forms an equivalence relation and each equivalence class separates two connected subgraphs of the graph from each other. A Hamming labeling may be obtained by assigning one bit of each label to each of the equivalence classes of the Djoković–Winkler relation; in one of the two connected subgraphs separated by an equivalence class of edges, all of the vertices have a 0 in that position of their labels, and in the other connected subgraph all of the vertices have a 1 in the same position.

67.4 Recognition

Partial cubes can be recognized, and a Hamming labeling constructed, in O(n2) time, where n is the number of vertices in the graph.[8] Given a partial cube, it is straightforward to construct the equivalence classes of the Djoković– Winkler relation by doing a breadth ﬁrst search from each vertex, in total time O(nm) ; the O(n2) -time recognition algorithm speeds this up by using bit-level parallelism to perform multiple breadth ﬁrst searches in a single pass 67.5. DIMENSION 245 through the graph, and then applies a separate algorithm to verify that the result of this computation is a valid partial cube labeling.

67.5 Dimension

The isometric dimension of a partial cube is the minimum dimension of a hypercube onto which it may be isometrically embedded, and is equal to the number of equivalence classes of the Djoković–Winkler relation. For instance, the isometric dimension of an n -vertex tree is its number of edges, n − 1 . An embedding of a partial cube onto a hypercube of this dimension is unique, up to symmetries of the hypercube.[9] Every hypercube and therefore every partial cube may be embedded isometrically into an integer lattice, and the lattice dimension of the partial cube is the minimum dimension of an integer lattice for which this is possible. The lattice dimension may be signiﬁcantly smaller than the isometric dimension; for instance, for a tree it is half the number of leaves in the tree (rounded up to the nearest integer). The lattice dimension of any graph, and a lattice embedding of minimum dimension, may be found in polynomial time by an algorithm based on maximum matching in an auxiliary graph.[10] Other types of dimension of partial cubes have also been deﬁned, based on embeddings into more specialized structures.[11]

67.6 Application to chemical graph theory

Isometric embeddings of graphs into hypercubes have an important application in chemical graph theory.A benzenoid graph is a graph consisting of all vertices and edges lying on and in the interior of a cycle in a hexagonal lattice. Such graphs are the molecular graphs of the benzenoid hydrocarbons, a large class of organic molecules. Every such graph is a partial cube. A Hamming labeling of such a graph can be used to compute the Wiener index of the corresponding molecule, which can then be used to predict certain of its chemical properties.[12] A diﬀerent molecular structure formed from carbon, the diamond cubic, also forms partial cube graphs.[13]

67.7 Notes

[1] Ovchinnikov (2011), Deﬁnition 5.1, p. 127.

[2] Ovchinnikov (2011), p. 174.

[3] Ovchinnikov (2011), Section 5.11, “Median Graphs”, pp. 163–165.

[4] Ovchinnikov (2011), Chapter 7, “Hyperplane Arrangements”, pp. 207–235.

[5] Eppstein (2006).

[6] Ovchinnikov (2011), Section 5.7, “Cartesian Products of Partial Cubes”, pp. 144–145.

[7] Winkler (1984), Theorem 4. See also Ovchinnikov (2011), Deﬁnition 2.13, p.29, and Theorem 5.19, p. 136.

[8] Eppstein (2008).

[9] Ovchinnikov (2011), Section 5.6, “Isometric Dimension”, pp. 142–144, and Section 5.10, “Uniqueness of Isometric Em- beddings”, pp. 157–162.

[10] Hadlock & Hoﬀman (1978); Eppstein (2005); Ovchinnikov (2011), Chapter 6, “Lattice Embeddings”, pp. 183–205.

[11] Eppstein (2009); Cabello, Eppstein & Klavžar (2011).

[12] Klavžar, Gutman & Mohar (1995), Propositions 2.1 and 3.1; Imrich & Klavžar (2000), p. 60; Ovchinnikov (2011), Section 5.12, “Average Length and the Wiener Index”, pp. 165–168.

[13] Eppstein (2009). 246 CHAPTER 67. PARTIAL CUBE

67.8 References

• Cabello, S.; Eppstein, D.; Klavžar, S. (2011), “The Fibonacci dimension of a graph”, Electronic Journal of Combinatorics 18 (1), P55, arXiv:0903.2507. • Djoković, D. Ž. (1973), “Distance-preserving subgraphs of hypercubes”, Journal of Combinatorial Theory, Series B 14 (3): 263–267, doi:10.1016/0095-8956(73)90010-5, MR 0314669.

• Eppstein, David (2005), “The lattice dimension of a graph”, European Journal of Combinatorics 26 (6): 585– 592, arXiv:cs.DS/0402028, doi:10.1016/j.ejc.2004.05.001.

• Eppstein, David (2006), “Cubic partial cubes from simplicial arrangements”, Electronic Journal of Combina- torics 13 (1), R79, arXiv:math.CO/0510263.

• Eppstein, David (2008), “Recognizing partial cubes in quadratic time”, Proc. 19th ACM-SIAM Symposium on Discrete Algorithms, pp. 1258–1266, arXiv:0705.1025.

• Eppstein, David (2009), “Isometric diamond subgraphs”, Proc. 16th International Symposium on Graph Draw- ing, Heraklion, Crete, 2008, Lecture Notes in Computer Science 5417, Springer-Verlag, pp. 384–389, arXiv:0807.2218, doi:10.1007/978-3-642-00219-9_37. • Falmagne, J.-C.; Doignon, J.-P. (1997), “Stochastic evolution of rationality”, Theory and Decision 43: 107– 138, doi:10.1023/A:1004981430688. • Firsov, V.V. (1965), “On isometric embedding of a graph into a Boolean cube”, Cybernetics 1: 112–113, doi:10.1007/bf01074705. As cited by Ovchinnikov (2011). • Hadlock, F.; Hoﬀman, F. (1978), “Manhattan trees”, Utilitas Mathematica 13: 55–67. As cited by Ovchinnikov (2011). • Imrich, Wilfried; Klavžar, Sandi (2000), Product Graphs: Structure and Recognition, Wiley-Interscience Series in Discrete Mathematics and Optimization, New York: John Wiley & Sons, ISBN 978-0-471-37039-0, MR 1788124.

• Klavžar, Sandi; Gutman, Ivan; Mohar, Bojan (1995), “Labeling of benzenoid systems which reﬂects the vertex-distance relations” (PDF), Journal of Chemical Information and Computer Sciences 35 (3): 590–593, doi:10.1021/ci00025a030. • Kuzmin, V.; Ovchinnikov, S. (1975), “Geometry of preferences spaces I”, Automation and Remote Control 36: 2059–2063. As cited by Ovchinnikov (2011).

• Ovchinnikov, Sergei (2011), Graphs and Cubes, Universitext, Springer. See especially Chapter 5, “Partial Cubes”, pp. 127–181.

• Winkler, Peter M. (1984), “Isometric embedding in products of complete graphs”, Discrete Applied Mathe- matics 7 (2): 221–225, doi:10.1016/0166-218X(84)90069-6, MR 0727925. Chapter 68

Partial k-tree

In graph theory, a partial k-tree is a type of graph, deﬁned either as a subgraph of a k-tree or as a graph with treewidth at most k. Many NP-hard combinatorial problems on graphs are solvable in polynomial time when restricted to the partial k-trees, for bounded values of k.

68.1 Graph minors

For any fixed constant k, the partial k-trees are closed under the operation of graph minors, and therefore, by the Robertson–Seymour theorem, this family can be characterized in terms of a finite set of forbidden minors. The partial 1-trees are exactly the forests, and their single forbidden minor is a triangle. For the partial 2-trees the single forbidden minor is the complete graph on four vertices. However, the number of forbidden minors increases for larger values of k. For partial 3-trees there are four forbidden minors: the complete graph on five vertices, the octahedral graph with six vertices, the eight-vertex Wagner graph, and the pentagonal prism with ten vertices.[1]

68.2 Dynamic programming

Many algorithmic problems that are NP-complete for arbitrary graphs may be solved eﬃciently for partial k-trees by dynamic programming, using the tree decompositions of these graphs.[2]

68.3 Related families of graphs

If a family of graphs has bounded treewidth, then it is a subfamily of the partial k-trees, where k is the bound on the treewidth. Families of graphs with this property include the cactus graphs, pseudoforests, series-parallel graphs, outerplanar graphs, Halin graphs, and Apollonian networks.[1] For instance, the series-parallel graphs are a subfamily of the partial 2-trees, and more strongly a graph is a partial 2-tree if and only if each of its biconnected components is series-parallel. The control ﬂow graphs arising in the compilation of structured programs also have bounded treewidth, which allows certain tasks such as register allocation to be performed eﬃciently on them.[3]

68.4 Notes

[1] Bodlaender (1998).

[2] Arnborg & Proskurowski (1989); Bern, Lawler & Wong (1987); Bodlaender (1988).

[3] Thorup (1998).

247 248 CHAPTER 68. PARTIAL K-TREE

Forbidden minors for partial 3-trees

68.5 References

• Arnborg, S.; Proskurowski, A. (1989), “Linear time algorithms for NP-hard problems restricted to partial k-trees”, Discrete Applied Mathematics 23 (1): 11–24, doi:10.1016/0166-218X(89)90031-0.

• Bern, M. W.; Lawler, E. L.; Wong, A. L. (1987), “Linear-time computation of optimal subgraphs of decomposable graphs”, Journal of Algorithms 8 (2): 216–235, doi:10.1016/0196-6774(87)90039-3.

• Bodlaender, Hans L. (1988), “Dynamic programming on graphs with bounded treewidth”, Proc. 15th Inter- national Colloquium on Automata, Languages and Programming, Lecture Notes in Computer Science 317, Springer-Verlag, pp. 105–118, doi:10.1007/3-540-19488-6_110. • Bodlaender, Hans L. (1998), “A partial k-arboretum of graphs with bounded treewidth”, Theoretical Computer Science 209 (1–2): 1–45, doi:10.1016/S0304-3975(97)00228-4. • Thorup, Mikkel (1998), “All structured programs have small tree width and good register allocation”, Infor- mation and Computation 142 (2): 159–181, doi:10.1006/inco.1997.2697. Chapter 69

Petersen family

The Petersen family. K6 is at the top of the illustration, and the Petersen graph is at the bottom. The blue links indicate Δ-Y or Y-Δ transforms between graphs in the family.

249 250 CHAPTER 69. PETERSEN FAMILY

In graph theory, the Petersen family is a set of seven undirected graphs that includes the Petersen graph and the complete graph K6. The Petersen family is named after Danish mathematician Julius Petersen, the namesake of the Petersen graph. Any of the graphs in the Petersen family can be transformed into any other graph in the family by Δ-Y or Y-Δ transforms, operations in which a triangle is replaced by a degree-three vertex or vice versa. These seven graphs form the forbidden minors for linklessly embeddable graphs, graphs that can be embedded into three-dimensional space in such a way that no two cycles in the graph are linked.[1] They are also among the forbidden minors for the YΔY-reducible graphs.[2][3]

69.1 Deﬁnition

The form of Δ-Y and Y-Δ transforms used to deﬁne the Petersen family is as follows:

• If a graph G contains a vertex v with exactly three neighbors, then the Y-Δ transform of G at v is the graph formed by removing v from G and adding edges between each pair of its three neighbors.

• If a graph H contains a triangle uvw, then the Δ-Y transform of H at uvw is the graph formed by removing edges uv, vw, and uw from H and adding a new vertex connected to all three of u, v, and w.

These transformations are so called because of the Δ shape of a triangle in a graph and the Y shape of a degree-three vertex. Although these operations can in principle lead to multigraphs, that does not happen within the Petersen family. Because these operations preserve the number of edges in a graph, there are only ﬁnitely many graphs or multigraphs that can be formed from a single starting graph G by combinations of Δ-Y and Y-Δ transforms. The Petersen family then consists of every graph that can be reached from the Petersen graph by a combination of Δ-Y and Y-Δ transforms. There are seven graphs in the family, including the complete graph K6 on six vertices, the eight-vertex graph formed by removing a single edge from the complete bipartite graph K₄,₄, and the seven-vertex complete tripartite graph K₃,₃,₁.

69.2 Forbidden minors

A minor of a graph G is another graph formed from G by contracting and removing edges. As the Robertson– Seymour theorem shows, many important families of graphs can be characterized by a ﬁnite set of forbidden minors: for instance, according to Wagner’s theorem, the planar graphs are exactly the graphs that have neither the complete graph K5 nor the complete bipartite graph K₃,₃ as minors. Neil Robertson, Paul Seymour, and Robin Thomas used the Petersen family as part of a similar characterization of linkless embeddings of graphs, embeddings of a given graph into Euclidean space in such a way that every cycle in the graph is the boundary of a disk that is not crossed by any other part of the graph.[1] Horst Sachs had previously studied such embeddings, shown that the seven graphs of the Petersen family do not have such embeddings, and posed the question of characterizing the linklessly embeddable graphs by forbidden subgraphs.[4] Robertson et al. solved Sachs’ question by showing that the linkless embeddable graphs are exactly the graphs that do not have a member of the Petersen family as a minor. The Petersen family also form some of the forbidden minors for another family of graphs, the YΔY-reducible graphs. A connected graph is YΔY-reducible if it can be reduced to a single vertex by a sequence of steps, each of which is a Δ-Y or Y-Δ transform, the removal of a self-loop or multiple adjacency, the removal of a vertex with one neighbor, and the replacement of a vertex of degree two and its two neighboring edges by a single edge. For instance, the complete graph K4 can be reduced to a single vertex by a Y-Δ transform that turns it into a triangle with doubled edges, removal of the three doubled edges, a Δ-Y transform that turns it into the claw K₁,₃, and removal of the three degree-one vertices of the claw. Each of the Petersen family graphs forms a minimal forbidden minor for the family of YΔY-reducible graphs.[2] However, Neil Robertson provided an example of an apex graph (a linkless embeddable graph formed by adding one vertex to a planar graph) that is not YΔY-reducible, showing that the YΔY-reducible graphs form a proper subclass of the linkless embeddable graphs and have additional forbidden minors.[2] In fact, as Yaming Yu showed, there are at least 68,897,913,652 forbidden minors for the YΔY-reducible graphs beyond the seven of the Petersen family.[3] 69.3. REFERENCES 251

Robertson’s irreducible apex graph, showing that the YΔY-reducible graphs have additional forbidden minors beyond those in the Petersen family.

69.3 References

[1] Robertson, Neil; Seymour, P. D.; Thomas, Robin (1993), “Linkless embeddings of graphs in 3-space”, Bulletin of the Amer- ican Mathematical Society 28 (1): 84–89, arXiv:math/9301216, doi:10.1090/S0273-0979-1993-00335-5, MR 1164063.

[2] Truemper, Klaus (1992), Matroid Decomposition (PDF), Academic Press, pp. 100–101.

[3] Yu, Yaming (2006), “More forbidden minors for wye-delta-wye reducibility” (PDF), Electronic Journal of Combinatorics 13 (1): #R7.

[4] Sachs, Horst (1983), “On a spatial analogue of Kuratowski’s Theorem on planar graphs – an open problem”, in Horowiecki, M.; Kennedy, J. W.; Sysło, M. M., Graph Theory: Proceedings of a Conference held in Łagów, Poland, February 10–13, 1981, Lecture Notes in Mathematics 1018, Springer-Verlag, pp. 230–241, doi:10.1007/BFb0071633. Chapter 70

Planar graph

“Triangular graph” redirects here. For triangulated graphs, see Chordal graph. For data graphs plotted across three variables, see Ternary plot.

In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other.[1] Such a drawing is called a plane graph or planar embedding of the graph. A plane graph can be deﬁned as a planar graph with a mapping from every node to a point on a plane, and from every edge to a plane curve on that plane, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points. Every graph that can be drawn on a plane can be drawn on the sphere as well, and vice versa. Plane graphs can be encoded by combinatorial maps. The equivalence class of topologically equivalent drawings on the sphere is called a planar map. Although a plane graph has an external or unbounded face, none of the faces of a planar map have a particular status. A generalization of planar graphs are graphs which can be drawn on a surface of a given genus. In this terminology, planar graphs have graph genus 0, since the plane (and the sphere) are surfaces of genus 0. See "graph embedding" for other related topics.

70.1 Kuratowski’s and Wagner’s theorems

The Polish mathematician Kazimierz Kuratowski provided a characterization of planar graphs in terms of forbidden graphs, now known as Kuratowski’s theorem:

A ﬁnite graph is planar if and only if it does not contain a subgraph that is a subdivision of K5 (the complete graph on ﬁve vertices) or K₃,₃ (complete bipartite graph on six vertices, three of which connect to each of the other three, also known as the utility graph).

A subdivision of a graph results from inserting vertices into edges (for example, changing an edge •——• to •—•—•) zero or more times. Instead of considering subdivisions, Wagner’s theorem deals with minors:

A ﬁnite graph is planar if and only if it does not have K5 or K₃,₃ as a minor.

Klaus Wagner asked more generally whether any minor-closed class of graphs is determined by a ﬁnite set of "forbidden minors". This is now the Robertson–Seymour theorem, proved in a long series of papers. In the language of this theorem, K5 and K₃,₃ are the forbidden minors for the class of ﬁnite planar graphs.

252 70.2. OTHER PLANARITY CRITERIA 253

A E

D G

An example of a graph which doesn't have K5 or K3,3 as its subgraph. However, it has a subgraph that is homeomorphic to K3,3 and is therefore not planar.

70.2 Other planarity criteria

In practice, it is diﬃcult to use Kuratowski’s criterion to quickly decide whether a given graph is planar. However, there exist fast algorithms for this problem: for a graph with n vertices, it is possible to determine in time O(n) (linear time) whether the graph may be planar or not (see planarity testing). For a simple, connected, planar graph with v vertices and e edges, the following simple conditions hold:

Theorem 1. If v ≥ 3 then e ≤ 3v − 6; Theorem 2. If v ≥ 3 and there are no cycles of length 3, then e ≤ 2v − 4.

In this sense, planar graphs are sparse graphs, in that they have only O(v) edges, asymptotically smaller than the maximum O(v2). The graph K₃,₃, for example, has 6 vertices, 9 edges, and no cycles of length 3. Therefore, by Theorem 2, it cannot be planar. Note that these theorems provide necessary conditions for planarity that are not suﬃcient conditions, and therefore can only be used to prove a graph is not planar, not that it is planar. If both theorem 1 and 2 fail, other methods may be used. 254 CHAPTER 70. PLANAR GRAPH

An animation showing that the Petersen graph contains a minor isomorphic to the K3,3 graph

• Whitney’s planarity criterion gives a characterization based on the existence of an algebraic dual; • Mac Lane’s planarity criterion gives an algebraic characterization of finite planar graphs, via their cycle spaces; • The Fraysseix–Rosenstiehl planarity criterion gives a characterization based on the existence of a bipartition of the cotree edges of a depth-first search tree. It is central to the left-right planarity testing algorithm; • Schnyder’s theorem gives a characterization of planarity in terms of partial order dimension; • Colin de Verdière’s planarity criterion gives a characterization based on the maximum multiplicity of the second eigenvalue of certain Schrödinger operators defined by the graph. • The Hanani–Tutte theorem states that a graph is planar if and only if it has a drawing in which each independent pair of edges crosses an even number of times; it can be used to characterize the planar graphs via a system of equations modulo 2.

70.2.1 Euler’s formula

Main article: Euler characteristic § Planar graphs

Euler’s formula states that if a ﬁnite, connected, planar graph is drawn in the plane without any edge intersections, and v is the number of vertices, e is the number of edges and f is the number of faces (regions bounded by edges, including the outer, inﬁnitely large region), then

v − e + f = 2

As an illustration, in the butterﬂy graph given above, v = 5, e = 6 and f = 3. If the second graph is redrawn without edge intersections, it has v = 4, e = 6 and f = 4. In general, if the property holds for all planar graphs of f faces, any change to the graph that creates an additional face while keeping the graph planar would keep v − e + f an invariant. Since the property holds for all graphs with f = 2, by mathematical induction it holds for all cases. Euler’s formula can also be proved as follows: if the graph isn't a tree, then remove an edge which completes a cycle. This lowers 70.2. OTHER PLANARITY CRITERIA 255

both e and f by one, leaving v − e + f constant. Repeat until the remaining graph is a tree; trees have v = e + 1 and f = 1, yielding v − e + f = 2. i.e. the Euler characteristic is 2. In a ﬁnite, connected, simple, planar graph, any face (except possibly the outer one) is bounded by at least three edges and every edge touches at most two faces; using Euler’s formula, one can then show that these graphs are sparse in the sense that if v ≥ 3:

e ≤ 3v − 6

Euler’s formula is also valid for convex polyhedra. This is no coincidence: every convex polyhedron can be turned

A Schlegel diagram of a regular dodecahedron, forming a planar graph from a convex polyhedron.

into a connected, simple, planar graph by using the Schlegel diagram of the polyhedron, a perspective projection of the polyhedron onto a plane with the center of perspective chosen near the center of one of the polyhedron’s faces. Not every planar graph corresponds to a convex polyhedron in this way: the trees do not, for example. Steinitz’s theorem says that the polyhedral graphs formed from convex polyhedra are precisely the ﬁnite 3-connected simple planar graphs. More generally, Euler’s formula applies to any polyhedron whose faces are simple polygons that form a surface topologically equivalent to a sphere, regardless of its convexity. 256 CHAPTER 70. PLANAR GRAPH

70.2.2 Average degree

From v − e + f = 2 and 2e ≥ 3f (one face has minimum 3 edges and each edge has maximum two faces) it follows via algebraic transformations that the average degree is strictly less than 6. Otherwise the given graph can't be planar.

70.2.3 Coin graphs

Main article: Circle packing theorem We say that two circles drawn in a plane kiss (or osculate) whenever they intersect in exactly one point. A “coin graph” is a graph formed by a set of circles, no two of which have overlapping interiors, by making a vertex for each circle and an edge for each pair of circles that kiss. The circle packing theorem, ﬁrst proved by Paul Koebe in 1936, states that a graph is planar if and only if it is a coin graph. This result provides an easy proof of Fáry’s theorem, that every planar graph can be embedded in the plane in such a way that its edges are straight line segments that do not cross each other. If one places each vertex of the graph at the center of the corresponding circle in a coin graph representation, then the line segments between centers of kissing circles do not cross any of the other edges.

70.3 Related families of graphs

70.3.1 Maximal planar graphs

A simple graph is called maximal planar if it is planar but adding any edge (on the given vertex set) would destroy that property. All faces (including the outer one) are then bounded by three edges, explaining the alternative term plane triangulation. The alternative names “triangular graph”[2] or “triangulated graph”[3] have also been used, but are ambiguous, as they more commonly refer to the line graph of a complete graph and to the chordal graphs respectively. Every maximal planar is 3-vertex-connected. If a maximal planar graph has v vertices with v > 2, then it has precisely 3v − 6 edges and 2v − 4 faces. Apollonian networks are the maximal planar graphs formed by repeatedly splitting triangular faces into triples of smaller triangles. Equivalently, they are the planar 3-trees. Strangulated graphs are the graphs in which every peripheral cycle is a triangle. In a maximal planar graph (or more generally a polyhedral graph) the peripheral cycles are the faces, so maximal planar graphs are strangulated. The strangulated graphs include also the chordal graphs, and are exactly the graphs that can be formed by clique-sums (without deleting edges) of complete graphs and maximal planar graphs.[4]

70.3.2 Outerplanar graphs

Outerplanar graphs are graphs with an embedding in the plane such that all vertices belong to the unbounded face of the embedding. Every outerplanar graph is planar, but the converse is not true: K4 is planar but not outerplanar. A theorem similar to Kuratowski’s states that a ﬁnite graph is outerplanar if and only if it does not contain a subdivision of K4 or of K₂,₃. A 1-outerplanar embedding of a graph is the same as an outerplanar embedding. For k > 1 a planar embedding is k-outerplanar if removing the vertices on the outer face results in a (k − 1)-outerplanar embedding. A graph is k-outerplanar if it has a k-outerplanar embedding.

70.3.3 Halin graphs

A Halin graph is a graph formed from an undirected plane tree (with no degree-two nodes) by connecting its leaves into a cycle, in the order given by the plane embedding of the tree. Equivalently, it is a polyhedral graph in which one face is adjacent to all the others. Every Halin graph is planar. Like outerplanar graphs, Halin graphs have low treewidth, making many algorithmic problems on them more easily solved than in unrestricted planar graphs.[5] 70.4. ENUMERATION OF PLANAR GRAPHS 257

70.3.4 Other related families

An apex graph is a graph that may be made planar by the removal of one vertex, and a k-apex graph is a graph that may be made planar by the removal of at most k vertices. A 1-planar graph is a graph that may be drawn in the plane with at most one simple crossing per edge, and a k-planar graph is a graph that may be drawn with at most k simple crossings per edge. A toroidal graph is a graph that can be embedded without crossings on the torus. More generally, the genus of a graph is the minimum genus of a two-dimensional surface into which the graph may be embedded; planar graphs have genus zero and nonplanar toroidal graphs have genus one. Any graph may be embedded into three-dimensional space without crossings. However, a three-dimensional analogue of the planar graphs is provided by the linklessly embeddable graphs, graphs that can be embedded into three- dimensional space in such a way that no two cycles are topologically linked with each other. In analogy to Kuratowski’s and Wagner’s characterizations of the planar graphs as being the graphs that do not contain K5 or K₃,₃ as a minor, the linklessly embeddable graphs may be characterized as the graphs that do not contain as a minor any of the seven graphs in the Petersen family. In analogy to the characterizations of the outerplanar and planar graphs as being the graphs with Colin de Verdière graph invariant at most two or three, the linklessly embeddable graphs are the graphs that have Colin de Verdière invariant at most four. An upward planar graph is a directed acyclic graph that can be drawn in the plane with its edges as non-crossing curves that are consistently oriented in an upward direction. Not every planar directed acyclic graph is upward planar, and it is NP-complete to test whether a given graph is upward planar.

70.4 Enumeration of planar graphs

The asymptotic for the number of (labeled) planar graphs on n vertices is g · n−7/2 · γn · n! , where γ ≈ 27.22687 and g ≈ 0.43 × 10−5 .[6] Almost all planar graphs have an exponential number of automorphisms.[7] The number of unlabeled (non-isomorphic) planar graphs on n vertices is between 27.2n and 30.06n .[8]

70.5 Other facts and deﬁnitions

Every planar graph is 4-partite, or 4-colorable; this is the graph-theoretical formulation of the four color theorem. Fáry’s theorem states that every simple planar graph admits an embedding in the plane such that all edges are straight line segments which don't intersect. A universal point set is a set of points such that every planar graph with n vertices has such an embedding with all vertices in the point set; there exist universal point sets of quadratic size, formed by taking a rectangular subset of the integer lattice. Every simple outerplanar graph admits an embedding in the plane such that all vertices lie on a fixed circle and all edges are straight line segments that lie inside the disk and don't intersect, so n-vertex regular polygons are universal for outerplanar graphs. Given an embedding G of a (not necessarily simple) connected graph in the plane without edge intersections, we construct the dual graph G* as follows: we choose one vertex in each face of G (including the outer face) and for each edge e in G we introduce a new edge in G* connecting the two vertices in G* corresponding to the two faces in G that meet at e. Furthermore, this edge is drawn so that it crosses e exactly once and that no other edge of G or G* is intersected. Then G* is again the embedding of a (not necessarily simple) planar graph; it has as many edges as G, as many vertices as G has faces and as many faces as G has vertices. The term “dual” is justified by the fact that G** = G; here the equality is the equivalence of embeddings on the sphere. If G is the planar graph corresponding to a convex polyhedron, then G* is the planar graph corresponding to the dual polyhedron. Duals are useful because many properties of the dual graph are related in simple ways to properties of the original graph, enabling results to be proven about graphs by examining their dual graphs. While the dual constructed for a particular embedding is unique (up to isomorphism), graphs may have different (i.e. non-isomorphic) duals, obtained from different (i.e. non-homeomorphic) embeddings. A Euclidean graph is a graph in which the vertices represent points in the plane, and the edges are assigned lengths equal to the Euclidean distance between those points; see Geometric graph theory. 258 CHAPTER 70. PLANAR GRAPH

A plane graph is said to be convex if all of its faces (including the outer face) are convex polygons. A planar graph may be drawn convexly if and only if it is a subdivision of a 3-vertex-connected planar graph. Scheinerman’s conjecture (now a theorem) states that every planar graph can be represented as an intersection graph of line segments in the plane. The planar separator theorem states that every n-vertex planar graph can be partitioned into two subgraphs of size at most 2n/3 by the removal of O(√n) vertices. As a consequence, planar graphs also have treewidth and branch-width O(√n). For two planar graphs with v vertices, it is possible to determine in time O(v) whether they are isomorphic or not (see also graph isomorphism problem).[9] The meshedness coeﬃcient of a planar graph normalizes its number of bounded faces (the same as the circuit rank of the graph, by Mac Lane’s planarity criterion) by dividing it by 2n − 5, the maximum possible number of bounded faces in a planar graph with n vertices. Thus, it ranges from 0 for trees to 1 for maximal planar graphs.[10]

70.6 See also

• Combinatorial map a combinatorial object that can encode plane graphs • Planarization, a planar graph formed from a drawing with crossings by replacing each crossing point by a new vertex • Thickness (graph theory), the smallest number of planar graphs into which the edges of a given graph may be partitioned • Planarity, a puzzle computer game in which the objective is to embed a planar graph onto a plane • Sprouts (game), a pencil-and-paper game where a planar graph subject to certain constraints is constructed as part of the game play • Three utilities problem, a popular puzzle

70.7 Notes

[1] Trudeau, Richard J. (1993). Introduction to Graph Theory (Corrected, enlarged republication. ed.). New York: Dover Pub. p. 64. ISBN 978-0-486-67870-2. Retrieved 8 August 2012. Thus a planar graph, when drawn on a flat surface, either has no edge-crossings or can be redrawn without them. [2] Schnyder, W. (1989), “Planar graphs and poset dimension”, Order 5: 323–343, doi:10.1007/BF00353652, MR 1010382. [3] Bhasker, Jayaram; Sahni, Sartaj (1988), “A linear algorithm to find a rectangular dual of a planar triangulated graph”, Algorithmica 3 (1–4): 247–278, doi:10.1007/BF01762117. [4] Seymour, P. D.; Weaver, R. W. (1984), “A generalization of chordal graphs”, Journal of Graph Theory 8 (2): 241–251, doi:10.1002/jgt.3190080206, MR 742878. [5] Sysło, Maciej M.; Proskurowski, Andrzej (1983), “On Halin graphs”, Graph Theory: Proceedings of a Conference held in Lagów, Poland, February 10–13, 1981, Lecture Notes in Mathematics 1018, Springer-Verlag, pp. 248–256, doi:10.1007/BFb0071635. [6] Giménez, Omer; Noy, Marc (2009). “Asymptotic enumeration and limit laws of planar graphs”. J. Amer. Math. Soc. 22: 309–329. doi:10.1090/s0894-0347-08-00624-3. [7] McDiarmid, Colin; Steger, Angelika; Welsh, Dominic J.A. “Random planar graphs”. Journal of Combinatorial Theory, Series B 93 (2): 187–205. doi:10.1016/j.jctb.2004.09.007. [8] Bonichon, N.; Gavoille, C.; Hanusse, N.; Poulalhon, D.; Schaeffer, G. (2006). “Planar Graphs, via Well-Orderly Maps and Trees”. Graphs and Combinatorics 22: 185–202. doi:10.1007/s00373-006-0647-2. [9] I. S. Filotti, Jack N. Mayer. A polynomial-time algorithm for determining the isomorphism of graphs of fixed genus. Proceedings of the 12th Annual ACM Symposium on Theory of Computing, p.236–243. 1980. [10] Buhl, J.; Gautrais, J.; Sole, R.V.; Kuntz, P.; Valverde, S.; Deneubourg, J.L.; Theraulaz, G. (2004), “Efficiency and robustness in ant networks of galleries”, The European Physical Journal B-Condensed Matter and Complex Systems (Springer- Verlag) 42 (1): 123–129, doi:10.1140/epjb/e2004-00364-9. 70.8. REFERENCES 259

70.8 References

• Kuratowski, Kazimierz (1930), “Sur le problème des courbes gauches en topologie” (PDF), Fund. Math. (in French) 15: 271–283. • Wagner, K. (1937), "Über eine Eigenschaft der ebenen Komplexe”, Math. Ann. 114: 570–590, doi:10.1007/BF01594196.

• Boyer, John M.; Myrvold, Wendy J. (2005), “On the cutting edge: Simpliﬁed O(n) planarity by edge addition” (PDF), Journal of Graph Algorithms and Applications 8 (3): 241–273, doi:10.7155/jgaa.00091. • McKay, Brendan; Brinkmann, Gunnar, A useful planar graph generator.

• de Fraysseix, H.; Ossona de Mendez, P.; Rosenstiehl, P. (2006), “Trémaux trees and planarity”, International Journal of Foundations of Computer Science 17 (5): 1017–1029, doi:10.1142/S0129054106004248. Special Issue on Graph Drawing. • D.A. Bader and S. Sreshta, A New Parallel Algorithm for Planarity Testing, UNM-ECE Technical Report 03-002, October 1, 2003. • Fisk, Steve (1978), “A short proof of Chvátal’s watchman theorem”, J. Comb. Theory, Ser. B 24 (3): 374, doi:10.1016/0095-8956(78)90059-X.

70.9 External links

• Edge Addition Planarity Algorithm Source Code, version 1.0 — Free C source code for reference implementation of Boyer–Myrvold planarity algorithm, which provides both a combinatorial planar embedder and Kuratowski subgraph isolator. An open source project with free licensing provides the Edge Addition Planarity Algorithms, current version. • Public Implementation of a Graph Algorithm Library and Editor — GPL graph algorithm library including planarity testing, planarity embedder and Kuratowski subgraph exhibition in linear time. • Boost Graph Library tools for planar graphs, including linear time planarity testing, embedding, Kuratowski subgraph isolation, and straight-line drawing.

• 3 Utilities Puzzle and Planar Graphs • NetLogo Planarity model — NetLogo version of John Tantalo’s game 260 CHAPTER 70. PLANAR GRAPH

− Example of the circle packing theorem on K 5, the complete graph on ﬁve vertices, minus one edge. 70.9. EXTERNAL LINKS 261

The Goldner–Harary graph is maximal planar. All its faces are bounded by three edges. 262 CHAPTER 70. PLANAR GRAPH

A planar graph and its dual Chapter 71

Platonic graph

Tetrahedron Octahedron Hexahedron

Square pyramid Icosahedron Dodecahedron

The platonic graphs can be seen as Schlegel diagrams of the platonic solids. (excluding the square pyramid also shown here)

In the mathematical ﬁeld of graph theory, a Platonic graph is a graph that has one of the Platonic solids as its skeleton. There are 5 Platonic graphs, and all of them are regular, polyhedral (and therefore by necessity also 3- vertex-connected, vertex-transitive, edge-transitive and planar graphs), and also Hamiltonian graphs.[1]

• Tetrahedral graph – 4 vertices, 6 edges

• Octahedral graph – 6 vertices, 12 edges

• Cubical graph – 8 vertices, 12 edges

• Icosahedral graph – 12 vertices, 30 edges

• Dodecahedral graph – 20 vertices, 30 edges

263 264 CHAPTER 71. PLATONIC GRAPH

71.1 See also

• Regular map (graph theory)

• Archimedean graph • Wheel graph

71.2 References

[1] Read, R. C. and Wilson, R. J. An Atlas of Graphs, Oxford, England: Oxford University Press, 2004 reprint, Chapter 6 special graphs pp. 261, 266.

71.3 External links

• Weisstein, Eric W., “Platonic graph”, MathWorld. Chapter 72

Polytope graph

A 3-dimensional convex polytope

A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn.[1] Some authors use the terms “convex polytope” and “convex polyhedron” interchangeably, while others prefer to draw a distinction between the notions of a polyhedron and a polytope. In addition, some texts require a polytope to be a bounded set, while others[2] (including this article) allow polytopes to be unbounded. The terms “bounded/unbounded convex polytope” will be used below whenever the boundedness

265 266 CHAPTER 72. POLYTOPE GRAPH

is critical to the discussed issue. Yet other texts treat a convex n-polytope as a surface or (n−1)-manifold. Convex polytopes play an important role both in various branches of mathematics and in applied areas, most notably in linear programming. A comprehensive and inﬂuential book in the subject, called Convex Polytopes, was published in 1967 by Branko Grünbaum. In 2003 the 2nd edition of the book was published, with signiﬁcant additional material contributed by new writers.[1] In Grünbaum’s book, and in some other texts in discrete geometry, convex polytopes are often simply called “polytopes”. Grünbaum points out that this is solely to avoid the endless repetition of the word “convex”, and that the discussion should throughout be understood as applying only to the convex variety. A polytope is called full-dimensional if it is an n-dimensional object in Rn.

72.1 Examples

• Many examples of bounded convex polytopes can be found in the article "polyhedron". • In the 2-dimensional case the full-dimensional examples are a half-plane, a strip between two parallel lines, an angle shape (the intersection of two non-parallel half-planes), a shape defined by a convex polygonal chain with two rays attached to its ends, and a convex polygon. • Special cases of an unbounded convex polytope are a slab between two parallel hyperplanes, a wedge defined by two non-parallel half-spaces, a polyhedral cylinder (infinite prism), and a polyhedral cone (infinite cone) defined by three or more half-spaces passing through a common point.

72.2 Deﬁnitions

A convex polytope may be defined in a number of ways, depending on what is more suitable for the problem at hand. Grünbaum’s definition is in terms of a convex set of points in space. Other important definitions are: as the intersection of half-spaces (half-space representation) and as the convex hull of a set of points (vertex representation).

72.2.1 Vertex representation (Convex hull)

In his book Convex polytopes, Grünbaum deﬁnes a convex polytope as a compact convex set with a ﬁnite number of extreme points:

A set K of Rn is convex if, for each pair of distinct points a, b in K, the closed segment with endpoints a and b is contained within K.

This is equivalent to defining a bounded convex polytope as the convex hull of a finite set of points, where the finite set must contain the set of extreme points of the polytope. Such a definition is called a vertex representation (V- representation or V-description).[1] For a compact convex polytope, the minimal V-description is unique and it is given by the set of the vertices of the polytope.[1]

72.2.2 Intersection of half-spaces

A convex polytope may be defined as an intersection of a finite number of half-spaces. Such definition is called a half-space representation (H-representation or H-description).[1] There exist infinitely many H-descriptions of a convex polytope. However, for a full-dimensional convex polytope, the minimal H-description is in fact unique and is given by the set of the facet-defining halfspaces.[1] A closed half-space can be written as a linear inequality:[1]

a1x1 + a2x2 + ··· + anxn ≤ b 72.3. PROPERTIES 267 where n is the dimension of the space containing the polytope under consideration. Hence, a closed convex polytope may be regarded as the set of solutions to the system of linear inequalities:

a11x1 + a12x2 + ··· + a1nxn ≤ b1 a21x1 + a22x2 + ··· + a2nxn ≤ b2 ...... am1x1 + am2x2 + ··· + amnxn ≤ bm where m is the number of half-spaces deﬁning the polytope. This can be concisely written as the matrix inequality:

Ax ≤ b where A is an m×n matrix, x is an n×1 column vector of variables, and b is an m×1 column vector of constants. An open convex polytope is defined in the same way, with strict inequalities used in the formulas instead of the non-strict ones. The coefficients of each row of A and b correspond with the coefficients of the linear inequality defining the respective half-space. Hence, each row in the matrix corresponds with a supporting hyperplane of the polytope, a hyperplane bounding a half-space that contains the polytope. If a supporting hyperplane also intersects the polytope, it is called a bounding hyperplane (since it is a supporting hyperplane, it can only intersect the polytope at the polytope’s boundary). The foregoing definition assumes that the polytope is full-dimensional. If it is not, then the solution of Ax ≤ b lies in a proper affine subspace of Rn and discussion of the polytope may be constrained to this subspace. In general the intersection of arbitrary half-spaces need not be bounded. However if one wishes to have a definition equivalent to that as a convex hull, then bounding must be explicitly required.

Finite basis theorem

The finite basis theorem[2] is an extension of the notion of V-description to include infinite polytopes. The theorem states that a convex polyhedron is the convex sum of its vertices plus the conical sum of the direction vectors of its infinite edges.

72.3 Properties

Every (bounded) convex polytope is the image of a simplex, as every point is a convex combination of the (finitely many) vertices. However, polytopes are not in general isomorphic to simplices. This is in contrast to the case of vector spaces and linear combinations, every finite-dimensional vector space being not only an image of, but in fact isomorphic to, Euclidean space of some dimension (or analog over other fields).

72.3.1 The face lattice

A face of a convex polytope is any intersection of the polytope with a halfspace such that none of the interior points of the polytope lie on the boundary of the halfspace. If a polytope is d-dimensional, its facets are its (d − 1)-dimensional faces, its vertices are its 0-dimensional faces, its edges are its 1-dimensional faces, and its ridges are its (d − 2)- dimensional faces. Given a convex polytope P deﬁned by the matrix inequality Ax ≤ b , if each row in A corresponds with a bounding hyperplane and is linearly independent of the other rows, then each facet of P corresponds with exactly one row of A, and vice versa. Each point on a given facet will satisfy the linear equality of the corresponding row in the matrix. (It may or may not also satisfy equality in other rows). Similarly, each point on a ridge will satisfy equality in two of the rows of A. 268 CHAPTER 72. POLYTOPE GRAPH

The face lattice of a square pyramid, drawn as a Hasse diagram; each face in the lattice is labeled by its vertex set.

In general, an (n − j)-dimensional face satisfies equality in j specific rows of A. These rows form a basis of the face. Geometrically speaking, this means that the face is the set of points on the polytope that lie in the intersection of j of the polytope’s bounding hyperplanes. The faces of a convex polytope thus form an Eulerian lattice called its face lattice, where the partial ordering is by set containment of faces. The definition of a face given above allows both the polytope itself and the empty set to be considered as faces, ensuring that every pair of faces has a join and a meet in the face lattice. The whole polytope is the unique maximum element of the lattice, and the empty set, considered to be a (−1)-dimensional face (a null polytope) of every polytope, is the unique minimum element of the lattice. Two polytopes are called combinatorially isomorphic if their face lattices are isomorphic. The polytope graph (polytopal graph, graph of the polytope, 1-skeleton) is the set of vertices and edges of the polytope only, ignoring higher-dimensional faces. For instance, a polyhedral graph is the polytope graph of a three- dimensional polytope. By a result of Whitney[3] the face lattice of a three-dimensional polytope is determined by its graph. The same is true for simple polytopes of arbitrary dimension (Blind & Mani-Levitska 1987, proving a conjecture of Micha Perles).[4] Kalai (1988)[5] gives a simple proof based on unique sink orientations. Because these polytopes’ face lattices are determined by their graphs, the problem of deciding whether two three-dimensional or simple convex polytopes are combinatorially isomorphic can be formulated equivalently as a special case of the graph isomorphism problem. However, it is also possible to translate these problems in the opposite direction, showing that polytope isomorphism testing is graph-isomorphism complete.[6]

72.3.2 Topological properties

A convex polytope, like any closed convex subset of Rn, is homeomorphic to a closed ball.[7] Let m denote the dimension of the polytope. If the polytope is full-dimensional, then m = n. The convex polytope therefore is an m- dimensional manifold with boundary, its Euler characteristic is 1, and its fundamental group is trivial. The boundary of the convex polytope is homeomorphic to an (m − 1)-sphere. The boundary’s Euler characteristic is 0 for even m and 2 for odd m. The boundary may also be regarded as a tessellation of (m − 1)-dimensional spherical space — i.e. as a spherical tiling.

72.3.3 Simplicial decomposition

A convex polytope can be decomposed into a simplicial complex, or union of simplices, satisfying certain properties. Given a convex r-dimensional polytope P, a subset of its vertices containing (r+1) aﬃnely independent points deﬁnes an r-simplex. It is possible to form a collection of subsets such that the union of the corresponding simplices is 72.4. ALGORITHMIC PROBLEMS FOR A CONVEX POLYTOPE 269 equal to P, and the intersection of any two simplices is either empty or a lower-dimensional simplex. This simplicial decomposition is the basis of many methods for computing the volume of a convex polytope, since the volume of a simplex is easily given by a formula.[8]

72.4 Algorithmic problems for a convex polytope

72.4.1 Construction of representations

Different representations of a convex polytope have different utility, therefore the construction of one representation given another one is an important problem. The problem of the construction of a V-representation is known as the vertex enumeration problem and the problem of the construction of a H-representation is known as the facet enumeration problem. While the vertex set of a bounded convex polytope uniquely defines it, in various applications it is important to know more about the combinatorial structure of the polytope, i.e., about its face lattice. Various convex hull algorithms deal both with the facet enumeration and face lattice construction. In the planar case, i.e., for a convex polygon, both facet and vertex enumeration problems amount to the ordering vertices (resp. edges) around the convex hull. It is a trivial task when the convex polygon is specified in a traditional for polygons way, i.e., by the ordered sequence of its vertices v1, . . . , vm . When the input list of vertices (or edges) is unordered, the time complexity of the problems becomes O(m log m).[9] A matching lower bound is known in the algebraic decision tree model of computation.[10]

72.5 See also

• Oriented matroid

• Nef polyhedron

72.6 References

[1] Branko Grünbaum, Convex Polytopes, 2nd edition, prepared by Volker Kaibel, Victor Klee, and Günter M. Ziegler, 2003, ISBN 0-387-40409-0, ISBN 978-0-387-40409-7, 466pp.

[2] Mathematical Programming, by Melvyn W. Jeter (1986) ISBN 0-8247-7478-7, p. 68

[3] Whitney, Hassler (1932). “Congruent graphs and the connectivity of graphs”. Amer. J. Math. 54 (1): 150–168. doi:10.2307/2371086. JSTOR 2371086.

[4] Blind, Roswitha; Mani-Levitska, Peter (1987), “Puzzles and polytope isomorphisms”, Aequationes Mathematicae 34 (2-3): 287–297, doi:10.1007/BF01830678, MR 921106.

[5] Kalai, Gil (1988), “A simple way to tell a simple polytope from its graph”, Journal of Combinatorial Theory, Ser. A 49 (2): 381–383, doi:10.1016/0097-3165(88)90064-7, MR 964396.

[6] Kaibel, Volker; Schwartz, Alexander (2003). “On the Complexity of Polytope Isomorphism Problems”. Graphs and Combinatorics 19 (2): 215–230. arXiv:math/0106093. doi:10.1007/s00373-002-0503-y.

[7] Glen E. Bredon, Topology and Geometry, 1993, ISBN 0-387-97926-3, p. 56.

[8] Büeler, B.; Enge, A.; Fukuda, K. (2000). “Exact Volume Computation for Polytopes: A Practical Study”. Polytopes — Combinatorics and Computation. p. 131. doi:10.1007/978-3-0348-8438-9_6. ISBN 978-3-7643-6351-2.

[9] Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Cliﬀord (2001) [1990]. “33.3 Finding the convex hull”. Introduction to Algorithms (2nd ed.). MIT Press and McGraw-Hill. pp. 947–957. ISBN 0-262-03293-7.

[10] Yao, Andrew Chi Chih (1981), “A lower bound to ﬁnding convex hulls”, Journal of the ACM 28 (4): 780–787, doi:10.1145/322276.322289, MR 677089; Ben-Or, Michael (1983), “Lower Bounds for Algebraic Computation Trees”, Proceedings of the Fifteenth An- nual ACM Symposium on Theory of Computing (STOC '83), pp. 80–86, doi:10.1145/800061.808735. 270 CHAPTER 72. POLYTOPE GRAPH

72.7 External links

• Weisstein, Eric W., “Convex polygon”, MathWorld.

• Weisstein, Eric W., “Convex polyhedron”, MathWorld. • Komei Fukuda, Polyhedral computation FAQ. Chapter 73

Prism graph

In the mathematical ﬁeld of graph theory, a prism graph is a graph that has one of the prisms as its skeleton.

73.1 Examples

The individual graphs may be named after the associated solid:

• Triangular prism graph – 6 vertices, 9 edges

• Cubical graph – 8 vertices, 12 edges

• Pentagonal prism graph – 10 vertices, 15 edges

• Hexagonal prism graph – 12 vertices, 18 edges

• Heptagonal prism graph – 14 vertices, 21 edges

• Octagonal prism graph – 16 vertices, 24 edges

• ...

Although geometrically the star polygons also form the faces of a diﬀerent sequence of (self-intersecting and nonconvex) prismatic polyhedra, the graphs of these star prisms are isomorphic to the prism graphs, and do not form a separate sequence of graphs.

73.2 Construction

Prism graphs are examples of generalized Petersen graphs, with parameters GP(n,1). They may also be constructed as the Cartesian product of a cycle graph with a single edge.[1] As with many vertex-transitive graphs, the prism graphs may also be constructed as Cayley graphs. The order-n dihedral group is the group of symmetries of a regular n-gon in the plane; it acts on the n-gon by rotations and reflections. It can be generated by two elements, a rotation by an angle of 2π/n and a single rotation, and its Cayley graph with this generating set is the prism graph. Abstractly, the group has the presentation ⟨r, f | rn, f 2, (rf)2⟩ (where r is a rotation and f is a reflection or flip) and the Cayley graph has r and f (or r, r−1, and f) as its generators.[1] 2,n The n-gonal prism graphs with odd values of n may be constructed as circulant graphs C2n . However, this construction does not work for even values of n.[1]

271 272 CHAPTER 73. PRISM GRAPH

73.3 Properties

The graph of an n-gonal prism has 2n vertices and 3n edges. They are regular, cubic graphs. Since the prism has symmetries taking each vertex to each other vertex, the prism graphs are vertex-transitive graphs. As polyhedral graphs, they are also 3-vertex-connected planar graphs. Every prism graph has a Hamiltonian cycle.[2] Among all biconnected cubic graphs, the prism graphs have within a constant factor of the largest possible number of 1-factorizations. A 1-factorization is a partition of the edge set of the graph into three perfect matchings, or equivalently an edge coloring of the graph with three colors. Every biconnected n-vertex cubic graph has O(2n/2) 1-factorizations, and the prism graphs have Ω(2n/2) 1-factorizations.[3] The number of spanning trees of an n-gonal prism graph is given by the formula[4]

n( √ √ (2 + 3)n + (2 − 3)n) 2 For n = 3, 4, 5, ... these numbers are

78, 388, 1810, 8106, 35294, ...

The n-gonal prism graphs for even values of n are partial cubes. They form one of the few known inﬁnite families of cubic partial cubes, and (except for four sporadic examples) the only vertex-transitive cubic partial cubes.[5] The pentagonal prism is one of the forbidden minors for the graphs of treewidth three.[6] The triangular prism and cube graph have treewidth exactly three, but all larger prism graphs have treewidth four.

73.4 Related graphs

Other inﬁnite sequences of polyhedral graph formed in a similar way from polyhedra with regular-polygon bases include the antiprism graphs (graphs of antiprisms) and wheel graphs (graphs of pyramids). Other vertex-transitive polyhedral graphs include the Archimedean graphs. If the two cycles of a prism graph are broken by the removal of a single edge in the same position in both cycles, the result is a ladder graph. If these two removed edges are replaced by two crossed edges, the result is a non-planar graph called a Möbius ladder.[7]

73.5 References

[1] Weisstein, Eric W., “Prism graph”, MathWorld.

[2] Read, R. C. and Wilson, R. J. An Atlas of Graphs, Oxford, England: Oxford University Press, 2004 reprint, Chapter 6 special graphs pp. 261, 270.

[3] Eppstein, David (2013), “The complexity of bendless three-dimensional orthogonal graph drawing”, Journal of Graph Algorithms and Applications 17 (1): 35–55, doi:10.7155/jgaa.00283, MR 3019198. Eppstein credits the observation that prism graphs have close to the maximum number of 1-factorizations to a personal communication by Greg Kuperberg.

[4] Jagers, A. A. (1988), “A note on the number of spanning trees in a prism graph”, International Journal of Computer Mathematics 24 (2): 151–154, doi:10.1080/00207168808803639.

[5] Marc, Tilen (2015), Classiﬁcation of vertex-transitive cubic partial cubes, arXiv:1509.04565.

[6] Arnborg, Stefan; Proskurowski, Andrzej; Corneil, Derek G. (1990), “Forbidden minors characterization of partial 3-trees”, Discrete Mathematics 80 (1): 1–19, doi:10.1016/0012-365X(90)90292-P, MR 1045920.

[7] Guy, Richard K.; Harary, Frank (1967), “On the Möbius ladders”, Canadian Mathematical Bulletin 10: 493–496, doi:10.4153/CMB- 1967-046-4, MR 0224499. Chapter 74

Pseudoforest

A 1-forest (a maximal pseudoforest), formed by three 1-trees

In graph theory, a pseudoforest is an undirected graph[1] in which every connected component has at most one cycle. That is, it is a system of vertices and edges connecting pairs of vertices, such that no two cycles of consecutive edges share any vertex with each other, nor can any two cycles be connected to each other by a path of consecutive edges. A pseudotree is a connected pseudoforest. The names are justiﬁed by analogy to the more commonly studied trees and forests. (A tree is a connected graph with no cycles; a forest is a disjoint union of trees.) Gabow and Tarjan[2] attribute the study of pseudoforests to Dantzig’s 1963 book on linear programming, in which pseudoforests arise in the solution of certain network ﬂow

273 274 CHAPTER 74. PSEUDOFOREST problems.[3] Pseudoforests also form graph-theoretic models of functions and occur in several algorithmic problems. Pseudoforests are sparse graphs – they have very few edges relative to their number of vertices – and their matroid structure allows several other families of sparse graphs to be decomposed as unions of forests and pseudoforests. The name “pseudoforest” comes from Picard & Queyranne (1982).

74.1 Deﬁnitions and structure

We define an undirected graph to be a set of vertices and edges such that each edge has two vertices (which may coincide) as endpoints. That is, we allow multiple edges (edges with the same pair of endpoints) and loops (edges whose two endpoints are the same vertex).[1] A subgraph of a graph is the graph formed by any subsets of its vertices and edges such that each edge in the edge subset has both endpoints in the vertex subset. A connected component of an undirected graph is the subgraph consisting of the vertices and edges that can be reached by following edges from a single given starting vertex. A graph is connected if every vertex or edge is reachable from every other vertex or edge. A cycle in an undirected graph is a connected subgraph in which each vertex is incident to exactly two edges, or is a loop.[4] A pseudoforest is an undirected graph in which each connected component contains at most one cycle.[5] Equivalently, it is an undirected graph in which each connected component has no more edges than vertices.[6] The components that have no cycles are just trees, while the components that have a single cycle within them are called 1-trees or unicyclic graphs. That is, a 1-tree is a connected graph containing exactly one cycle. A pseudoforest with a single connected component (usually called a pseudotree, although some authors define a pseudotree to be a 1-tree) is either a tree or a 1-tree; in general a pseudoforest may have multiple connected components as long as all of them are trees or 1-trees. If one removes from a 1-tree one of the edges in its cycle, the result is a tree. Reversing this process, if one augments a tree by connecting any two of its vertices by a new edge, the result is a 1-tree; the path in the tree connecting the two endpoints of the added edge, together with the added edge itself, form the 1-tree’s unique cycle. If one augments a 1-tree by adding an edge that connects one of its vertices to a newly added vertex, the result is again a 1-tree, with one more vertex; an alternative method for constructing 1-trees is to start with a single cycle and then repeat this augmentation operation any number of times. The edges of any 1-tree can be partitioned in a unique way into two subgraphs, one of which is a cycle and the other of which is a forest, such that each tree of the forest contains exactly one vertex of the cycle.[7] Certain more specific types of pseudoforests have also been studied.

A 1-forest, sometimes called a maximal pseudoforest, is a pseudoforest to which no more edges can be added without causing some component of the graph to contain multiple cycles. If a pseudoforest contains a tree as one of its components, it cannot be a 1-forest, for one can add either an edge connecting two vertices within that tree, forming a single cycle, or an edge connecting that tree to some other component. Thus, the 1-forests are exactly the pseudoforests in which every component is a 1-tree.

The spanning pseudoforests of an undirected graph G are the pseudoforest subgraphs of G that have all the vertices of G. Such a pseudoforest need not have any edges, since for example the subgraph that has all the vertices of G and no edges is a pseudoforest (whose components are trees consisting of a single vertex).

The maximal pseudoforests of G are the pseudoforest subgraphs of G that are not contained within any larger pseudoforest of G. A maximal pseudoforest of G is always a spanning pseudoforest, but not conversely. If G has no connected components that are trees, then its maximal pseudoforests are 1-forests, but if G does have a tree component, its maximal pseudoforests are not 1-forests. Stated precisely, in any graph G its maximal pseudoforests consist of every tree component of G, together with one or more disjoint 1-trees covering the remaining vertices of G.

74.2 Directed pseudoforests

Versions of these deﬁnitions are also used for directed graphs. Like an undirected graph, a directed graph consists of vertices and edges, but each edge is directed from one of its endpoints to the other endpoint. A directed pseudoforest 74.3. NUMBER OF EDGES 275

is a directed graph in which each vertex has at most one outgoing edge; that is, it has outdegree at most one. A directed 1-forest – most commonly called a functional graph (see below), sometimes maximal directed pseudoforest – is a directed graph in which each vertex has outdegree exactly one.[8] If D is a directed pseudoforest, the undirected graph formed by removing the direction from each edge of D is an undirected pseudoforest.

74.3 Number of edges

Every pseudoforest on a set of n vertices has at most n edges, and every maximal pseudoforest on a set of n vertices has exactly n edges. Conversely, if a graph G has the property that, for every subset S of its vertices, the number of edges in the induced subgraph of S is at most the number of vertices in S, then G is a pseudoforest. 1-trees can be defined as connected graphs with equally many vertices and edges.[2] Moving from individual graphs to graph families, if a family of graphs has the property that every subgraph of a graph in the family is also in the family, and every graph in the family has at most as many edges as vertices, then the family contains only pseudoforests. For instance, every subgraph of a thrackle (a graph drawn so that every pair of edges has one point of intersection) is also a thrackle, so Conway’s conjecture that every thrackle has at most as many edges as vertices can be restated as saying that every thrackle is a pseudoforest. A more precise characterization is that, if the conjecture is true, then the thrackles are exactly the pseudoforests with no four-vertex cycle and at most one odd cycle.[9] Streinu and Theran[10] generalize the sparsity conditions defining pseudoforests: they define a graph as being (k,l)- sparse if every nonempty subgraph with n vertices has at most kn − l edges, and (k,l)-tight if it is (k,l)-sparse and has exactly kn − l edges. Thus, the pseudoforests are the (1,0)-sparse graphs, and the maximal pseudoforests are the (1,0)-tight graphs. Several other important families of graphs may be defined from other values of k and l, and when l ≤ k the (k,l)-sparse graphs may be characterized as the graphs formed as the edge-disjoint union of l forests and k − l pseudoforests.[11] Almost every sufficiently sparse random graph is pseudoforest.[12] That is, if c is a constant with 0 < c < 1/2, and Pc(n) is the probability that choosing uniformly at random among the n-vertex graphs with cn edges results in a pseudoforest, then Pc(n) tends to one in the limit for large n. However, for c > 1/2, almost every random graph with cn edges has a large component that is not unicyclic.

74.4 Enumeration

A graph is simple if it has no self-loops and no multiple edges with the same endpoints. The number of simple 1-trees with n labelled vertices is[13]

( ) ( ) n ( ) ∑ (−1)k−1 ∑ n! n1 + ··· + nk n 2 2 . k n1! ··· nk! n k=1 n1+···+nk=n

The values for n up to 18 can be found in sequence A057500 of the On-Line Encyclopedia of Integer Sequences. The number of maximal directed pseudoforests on n vertices, allowing self-loops, is nn, because for each vertex there are n possible endpoints for the outgoing edge. André Joyal used this fact to provide a bijective proof of Cayley’s formula, that the number of undirected trees on n nodes is nn − 2, by ﬁnding a bijection between maximal directed pseudoforests and undirected trees with two distinguished nodes.[14] If self-loops are not allowed, the number of maximal directed pseudoforests is instead (n − 1)n.

74.5 Graphs of functions

“Functional graph” redirects here. For other uses, see Graph of a function. Directed pseudoforests and endofunctions are in some sense mathematically equivalent. Any function ƒ from a set X to itself (that is, an endomorphism of X) can be interpreted as deﬁning a directed pseudoforest which has an edge from x to y whenever ƒ(x) = y. The resulting directed pseudoforest is maximal, and may include self-loops whenever some value x has ƒ(x) = x. Alternatively, omitting the self-loops produces a non-maximal pseudoforest. In the other 276 CHAPTER 74. PSEUDOFOREST

direction, any maximal directed pseudoforest determines a function ƒ such that ƒ(x) is the target of the edge that goes out from x, and any non-maximal directed pseudoforest can be made maximal by adding self-loops and then converted into a function in the same way. For this reason, maximal directed pseudoforests are sometimes called functional graphs.[2] Viewing a function as a functional graph provides a convenient language for describing properties that are not as easily described from the function-theoretic point of view; this technique is especially applicable to problems involving iterated functions, which correspond to paths in functional graphs. Cycle detection, the problem of following a path in a functional graph to find a cycle in it, has applications in cryptography and computational number theory, as part of Pollard’s rho algorithm for integer factorization and as a method for finding collisions in cryptographic hash functions. In these applications, ƒ is expected to behave randomly; Flajolet and Odlyzko[15] study the graph-theoretic properties of the functional graphs arising from randomly chosen mappings. In particular, a form of the birthday paradox implies that, in a random functional graph with n vertices, the path starting from a randomly selected vertex will typically loop back on itself to form a cycle within O(√n) steps. Konyagin et. al. have made analytical and computational progress on graph statistics.[16] Martin, Odlyzko, and Wolfram[17] investigate pseudoforests that model the dynamics of cellular automata. These functional graphs, which they call state transition diagrams, have one vertex for each possible configuration that the ensemble of cells of the automaton can be in, and an edge connecting each configuration to the configuration that follows it according to the automaton’s rule. One can infer properties of the automaton from the structure of these diagrams, such as the number of components, length of limiting cycles, depth of the trees connecting non-limiting states to these cycles, or symmetries of the diagram. For instance, any vertex with no incoming edge corresponds to a Garden of Eden pattern and a vertex with a self-loop corresponds to a still life pattern. Another early application of functional graphs is in the trains used to study Steiner triple systems.[18] The train of a triple system is a functional graph having a vertex for each possible triple of symbols; each triple pqr is mapped by ƒ to stu, where pqs, prt, and qru are the triples that belong to the triple system and contain the pairs pq, pr, and qr respectively. Trains have been shown to be a powerful invariant of triple systems although somewhat cumbersome to compute.

74.6 Bicircular matroid

A matroid is a mathematical structure in which certain sets of elements are defined to be independent, in such a way that the independent sets satisfy properties modeled after the properties of linear independence in a vector space. One of the standard examples of a matroid is the graphic matroid in which the independent sets are the sets of edges in forests of a graph; the matroid structure of forests is important in algorithms for computing the minimum spanning tree of the graph. Analogously, we may define matroids from pseudoforests. For any graph G = (V,E), we may define a matroid on the edges of G, in which a set of edges is independent if and only if it forms a pseudoforest; this matroid is known as the bicircular matroid (or bicycle matroid) of G.[19][20] The smallest dependent sets for this matroid are the minimal connected subgraphs of G that have more than one cycle, and these subgraphs are sometimes called bicycles. There are three possible types of bicycle: a theta graph has two vertices that are connected by three internally disjoint paths, a figure 8 graph consists of two cycles sharing a single vertex, and a handcuff graph is formed by two disjoint cycles connected by a path.[21] A graph is a pseudoforest if and only if it does not contain a bicycle as a subgraph. [10]

74.7 Forbidden minors

Forming a minor of a pseudoforest by contracting some of its edges and deleting others produces another pseudoforest. Therefore, the family of pseudoforests is closed under minors, and the Robertson–Seymour theorem implies that pseudoforests can be characterized in terms of a finite set of forbidden minors, analogously to Wagner’s theorem characterizing the planar graphs as the graphs having neither the complete graph K5 nor the complete bipartite graph K₃,₃ as minors. As discussed above, any non-pseudoforest graph contains as a subgraph a handcuff, figure 8, or theta graph; any handcuff or figure 8 graph may be contracted to form a butterfly graph (five-vertex figure 8), and any theta graph may be contracted to form a diamond graph (four-vertex theta graph),[22] so any non-pseudoforest contains either a butterfly or a diamond as a minor, and these are the only minor-minimal non-pseudoforest graphs. Thus, a graph is a pseudoforest if and only if it does not have the butterfly or the diamond as a minor. If one forbids only the diamond but not the butterfly, the resulting larger graph family consists of the cactus graphs and disjoint unions of multiple cactus graphs.[23] 74.8. ALGORITHMS 277

More simply, if multigraphs with self-loops are considered, there is only one forbidden minor, a vertex with two loops.

74.8 Algorithms

An early algorithmic use of pseudoforests involves the network simplex algorithm and its application to generalized flow problems modeling the conversion between commodities of different types.[3][24] In these problems, one is given as input a flow network in which the vertices model each commodity and the edges model allowable conversions between one commodity and another. Each edge is marked with a capacity (how much of a commodity can be converted per unit time), a flow multiplier (the conversion rate between commodities), and a cost (how much loss or, if negative, profit is incurred per unit of conversion). The task is to determine how much of each commodity to convert via each edge of the flow network, in order to minimize cost or maximize profit, while obeying the capacity constraints and not allowing commodities of any type to accumulate unused. This type of problem can be formulated as a linear program, and solved using the simplex algorithm. The intermediate solutions arising from this algorithm, as well as the eventual optimal solution, have a special structure: each edge in the input network is either unused or used to its full capacity, except for a subset of the edges, forming a spanning pseudoforest of the input network, for which the flow amounts may lie between zero and the full capacity. In this application, unicyclic graphs are also sometimes called augmented trees and maximal pseudoforests are also sometimes called augmented forests.[24] The minimum spanning pseudoforest problem involves finding a spanning pseudoforest of minimum weight in a larger edge-weighted graph G. Due to the matroid structure of pseudoforests, minimum-weight maximal pseudoforests may be found by greedy algorithms similar to those for the minimum spanning tree problem. However, Gabow and Tarjan found a more efficient linear-time approach in this case.[2] The pseudoarboricity of a graph G is defined by analogy to the arboricity as the minimum number of pseudoforests into which its edges can be partitioned; equivalently, it is the minimum k such that G is (k,0)-sparse, or the minimum k such that the edges of G can be oriented to form a directed graph with outdegree at most k. Due to the matroid structure of pseudoforests, the pseudoarboricity may be computed in polynomial time.[25] A random bipartite graph with n vertices on each side of its bipartition, and with cn edges chosen independently at random from each of the n2 possible pairs of vertices, is a pseudoforest with high probability whenever c is a constant strictly less than one. This fact plays a key role in the analysis of cuckoo hashing, a data structure for looking up key-value pairs by looking in one of two hash tables at locations determined from the key: one can form a graph, the “cuckoo graph”, whose vertices correspond to hash table locations and whose edges link the two locations at which one of the keys might be found, and the cuckoo hashing algorithm succeeds in finding locations for all of its keys if and only if the cuckoo graph is a pseudoforest.[26] Pseudoforests also play a key role in parallel algorithms for graph coloring and related problems.[27]

74.9 Notes

[1] The kind of undirected graph considered here is often called a multigraph or pseudograph, to distinguish it from a simple graph.

[2] Gabow & Tarjan (1988).

[3] Dantzig (1963).

[4] See the linked articles and the references therein for these deﬁnitions.

[5] This is the deﬁnition used, e.g., by Gabow & Westermann (1992).

[6] This is the deﬁnition in Gabow & Tarjan (1988).

[7] See, e.g., the proof of Lemma 4 in Àlvarez, Blesa & Serna (2002).

[8] Kruskal, Rudolph & Snir (1990) instead use the opposite deﬁnition, in which each vertex has indegree one; the resulting graphs, which they call unicycular, are the transposes of the graphs considered here.

[9] Woodall (1969); Lovász, Pach & Szegedy (1997).

[10] Streinu & Theran (2009). 278 CHAPTER 74. PSEUDOFOREST

[11] Whiteley (1988).

[12] Bollobás (1985). See especially Corollary 24, p.120, for a bound on the number of vertices belonging to unicyclic components in a random graph, and Corollary 19, p.113, for a bound on the number of distinct labeled unicyclic graphs.

[13] Riddell (1951); see A057500 in the On-Line Encyclopedia of Integer Sequences.

[14] Aigner & Ziegler (1998).

[15] Flajolet & Odlyzko (1990).

[16] Konyagin et al. (2010).

[17] Martin, Odlyzko & Wolfram (1984).

[18] White (1913); Colbourn, Colbourn & Rosenbaum (1982); Stinson (1983).

[19] Simoes-Pereira (1972).

[20] Matthews (1977).

[21] Glossary of Signed and Gain Graphs and Allied Areas

[22] For this terminology, see the list of small graphs from the Information System on Graph Class Inclusions. However, butterfly graph may also refer to a different family of graphs related to hypercubes, and the five-vertex figure 8 is sometimes instead called a bowtie graph.

[23] El-Mallah & Colbourn (1988).

[24] Ahuja, Magnanti & Orlin (1993).

[25] Gabow & Westermann (1992). See also the faster approximation schemes of Kowalik (2006).

[26] Kutzelnigg (2006).

[27] Goldberg, Plotkin & Shannon (1988); Kruskal, Rudolph & Snir (1990).

74.10 References

• Ahuja, Ravindra K.; Magnanti, Thomas L.; Orlin, James B. (1993), Network Flows: Theory, Algorithms and Applications, Prentice Hall, ISBN 0-13-617549-X. • Aigner, Martin; Ziegler, Günter M. (1998), Proofs from THE BOOK, Springer-Verlag, pp. 141–146. • Àlvarez, Carme; Blesa, Maria; Serna, Maria (2002), “Universal stability of undirected graphs in the adver- sarial queueing model”, Proc. 14th ACM Symposium on Parallel Algorithms and Architectures, pp. 183–197, doi:10.1145/564870.564903. • Bollobás, Béla (1985), Random Graphs, Academic Press. • Colbourn, Marlene J.; Colbourn, Charles J.; Rosenbaum, Wilf L. (1982), “Trains: an invariant for Steiner triple systems”, Ars Combinatoria 13: 149–162, MR 0666934. • Dantzig, G. B. (1963), Linear Programming and Extensions, Princeton University Press. • El-Mallah, Ehab; Colbourn, Charles J. (1988), “The complexity of some edge deletion problems”, IEEE Trans- actions on Circuits and Systems 35 (3): 354–362, doi:10.1109/31.1748. • Flajolet, P.; Odlyzko, A. (1990), “Random mapping statistics”, Advances in Cryptology – EUROCRYPT '89: Workshop on the Theory and Application of Cryptographic Techniques, Lecture Notes in Computer Science 434, Springer-Verlag, pp. 329–354. • Gabow, H. N.; Tarjan, R. E. (1988), “A linear-time algorithm for ﬁnding a minimum spanning pseudoforest”, Information Processing Letters 27 (5): 259–263, doi:10.1016/0020-0190(88)90089-0. • Gabow, H. N.; Westermann, H. H. (1992), “Forests, frames, and games: Algorithms for matroid sums and applications”, Algorithmica 7 (1): 465–497, doi:10.1007/BF01758774. 74.11. EXTERNAL LINKS 279

• Goldberg, A. V.; Plotkin, S. A.; Shannon, G. E. (1988), “Parallel symmetry-breaking in sparse graphs”, SIAM Journal on Discrete Mathematics 1 (4): 434–446, doi:10.1137/0401044. • Konyagin, Sergei; Luca, Florian; Mans, Bernard; Mathieson, Luke; Shparlinski, Igor E. (2010), Functional Graphs of Polynomials over Finite Fields • Kowalik, Ł. (2006), “Approximation Scheme for Lowest Outdegree Orientation and Graph Density Measures”, in Asano, Tetsuo, Proceedings of the International Symposium on Algorithms and Computation, Lecture Notes in Computer Science 4288, Springer-Verlag, pp. 557–566, doi:10.1007/11940128.

• Kruskal, Clyde P.; Rudolph, Larry; Snir, Marc (1990), “Eﬃcient parallel algorithms for graph problems”, Algorithmica 5 (1): 43–64, doi:10.1007/BF01840376.

• Picard, Jean-Claude; Queyranne, Maurice (1982), “A network ﬂow solution to some nonlinear 0–1 programming problems, with applications to graph theory”, Networks 12 (2): 141–159, doi:10.1002/net.3230120206, MR 670021.

• Kutzelnigg, Reinhard (2006), “Bipartite random graphs and cuckoo hashing”, Fourth Colloquium on Mathe- matics and Computer Science, Discrete Mathematics and Theoretical Computer Science AG, pp. 403–406.

• Lovász, L.; Pach, J.; Szegedy, M. (1997), “On Conway’s thrackle conjecture”, Discrete and Computational Geometry 18 (4): 369–376, doi:10.1007/PL00009322.

• Martin, O.; Odlyzko, A. M.; Wolfram, S. (1984), “Algebraic properties of cellular automata”, Communications in Mathematical Physics 93 (2): 219–258, Bibcode:1984CMaPh..93..219M, doi:10.1007/BF01223745.

• Matthews, L. R. (1977), “Bicircular matroids”, The Quarterly Journal of Mathematics. Oxford. Second Series 28 (110): 213–227, doi:10.1093/qmath/28.2.213, MR 0505702.

• Riddell, R. J. (1951), Contributions to the Theory of Condensation, Ph.D. thesis, Ann Arbor: University of Michigan, Bibcode:1951PhDT...... 20R.

• Simoes-Pereira, J. M. S. (1972), “On subgraphs as matroid cells”, Mathematische Zeitschrift 127 (4): 315–322, doi:10.1007/BF01111390.

• Stinson, D. R. (1983), “A comparison of two invariants for Steiner triple systems: fragments and trains”, Ars Combinatoria 16: 69–76, MR doi=0734047.

• Streinu, I.; Theran, L. (2009), “Sparsity-certifying Graph Decompositions”, Graphs and Combinatorics 25 (2): 219, doi:10.1007/s00373-008-0834-4.

• White, H. S. (1913), “Triple-systems as transformations, and their paths among triads”, Transactions of the American Mathematical Society (American Mathematical Society) 14 (1): 6–13, doi:10.2307/1988765, JSTOR 1988765. • Whiteley, W. (1988), “The union of matroids and the rigidity of frameworks”, SIAM Journal on Discrete Mathematics 1 (2): 237–255, doi:10.1137/0401025. • Woodall, D. R. (1969), “Thrackles and deadlock”, in Welsh, D. J. A., Combinatorial Mathematics and Its Applications, Academic Press, pp. 335–348.

74.11 External links

• Weisstein, Eric W., “Unicyclic Graph”, MathWorld. 280 CHAPTER 74. PSEUDOFOREST

The 21 unicyclic graphs with at most six vertices 74.11. EXTERNAL LINKS 281

0 x ƒ(x) 8 1 0 6 1 6 2 0 7 2 3 1 4 4 5 3 6 6 3 3 7 4 5 8 0 4

A function from the set {0,1,2,3,4,5,6,7,8} to itself, and the corresponding functional graph

The butterﬂy graph (left) and diamond graph (right), forbidden minors for pseudoforests Chapter 75

Quartic graph

In the mathematical ﬁeld of graph theory, a quartic graph is a graph where all vertices have degree 4. In other words, a quartic graph is a 4-regular graph.[1]

75.1 Examples

Several well-known graphs are quartic. They include:

• The complete graph K5, a quartic graph with 5 vertices, the smallest possible quartic graph.

• The Chvátal graph, another quartic graph with 12 vertices, the smallest quartic graph that both has no triangles and cannot be colored with three colors.[2]

• The Folkman graph, a quartic graph with 20 vertices, the smallest semi-symmetric graph.[3]

• The Meredith graph, a quartic graph with 70 vertices that is 4-connected but has no Hamiltonian cycle, dis- proving a conjecture of Crispin Nash-Williams.[4]

Every medial graph is a quartic plane graph, and every quartic plane graph is the medial graph of a pair of dual plane graphs or multigraphs.[5] Knot diagrams and link diagrams are also quartic plane multigraphs, in which the vertices represent the crossings of the diagram and are marked with additional information concerning which of the two branches of the knot crosses the other branch at that point.[6]

75.2 Properties

Because the degree of every vertex in a quartic graph is even, every connected quartic graph has an Euler tour. And as with regular bipartite graphs more generally, every bipartite quartic graph has a perfect matching. In this case, a much simpler and faster algorithm for ﬁnding such a matching is possible than for irregular graphs: by selecting every other edge of an Euler tour, one may ﬁnd a 2-factor, which in this case must be a collection of cycles, each of even length, with each vertex of the graph appearing in exactly one cycle. By selecting every other edge again in these cycles, one obtains a perfect matching in linear time. The same method can also be used to color the edges of the graph with four colors in linear time.[7] Quartic graphs have an even number of Hamiltonian decompositions.[8]

75.3 Open problems

It is an open conjecture whether all quartic graphs have an even number of Hamiltonian circuits, or have more than one Hamiltonian circuit. The answer is known to be false for quartic multigraphs.[9]

282 75.4. SEE ALSO 283

The Chvátal graph

75.4 See also

• Cubic graph

75.5 References

[1] Toida, S. (1974), “Construction of quartic graphs”, Journal of Combinatorial Theory, Series B 16: 124–133, doi:10.1016/0095- 8956(74)90054-9, MR 0347693.

[2] Chvátal, V. (1970), “The smallest triangle-free 4-chromatic 4-regular graph”, Journal of Combinatorial Theory 9 (1): 93– 94, doi:10.1016/S0021-9800(70)80057-6.

[3] Folkman, Jon (1967), “Regular line-symmetric graphs”, Journal of Combinatorial Theory 3: 215–232, doi:10.1016/s0021- 9800(67)80069-3, MR 0224498.

[4] Meredith, G. H. J. (1973), “Regular n-valent n-connected nonHamiltonian non-n-edge-colorable graphs”, Journal of Com- binatorial Theory, Series B 14: 55–60, doi:10.1016/s0095-8956(73)80006-1, MR 0311503.

[5] Bondy, J. A.; Häggkvist, R. (1981), “Edge-disjoint Hamilton cycles in 4-regular planar graphs”, Aequationes Mathematicae 22 (1): 42–45, doi:10.1007/BF02190157, MR 623315. 284 CHAPTER 75. QUARTIC GRAPH

[6] Welsh, Dominic J. A. (1993), “The complexity of knots”, Quo vadis, graph theory?, Annals of Discrete Mathematics 55, Amsterdam: North-Holland, pp. 159–171, doi:10.1016/S0167-5060(08)70385-6, MR 1217989.

[7] Gabow, Harold N. (1976), “Using Euler partitions to edge color bipartite multigraphs”, International Journal of Computer and Information Sciences 5 (4): 345–355, doi:10.1007/bf00998632, MR 0422081.

[8] Thomason, A. G. (1978), “Hamiltonian cycles and uniquely edge colourable graphs”, Annals of Discrete Mathematics 3: 259–268, doi:10.1016/s0167-5060(08)70511-9, MR 499124.

[9] Fleischner, Herbert (1994), “Uniqueness of maximal dominating cycles in 3-regular graphs and of Hamiltonian cycles in 4-regular graphs”, Journal of Graph Theory 18 (5): 449–459, doi:10.1002/jgt.3190180503, MR 1283310.

75.6 External links

• Weisstein, Eric W., “Quartic Graph”, MathWorld. Chapter 76

Quasi-bipartite graph

In the mathematical ﬁeld of graph theory, an instance of the Steiner tree problem (consisting of an undirected graph G and a set R of terminal vertices that must be connected to each other) is said to be quasi-bipartite if the non-terminal vertices in G form an independent set, i.e. if every edge is incident on at least one terminal. This generalizes the concept of a bipartite graph: if G is bipartite, and R is the set of vertices on one side of the bipartition, the set to R is automatically independent. This concept was introduced by Rajagopalan and Vazirani [1] who used it to provide a (3/2 + ε) approximation algorithm for the Steiner tree problem on such instances. Subsequently the ε factor was removed by Rizzi[2] and a 4/3 approximation algorithm was obtained by Chakrabarty et al.[3] The same concept has been used by subsequent authors on the Steiner tree problem, e.g.[4] Robins and Zelikovsky[5] proposed an approximation algorithm for Steiner tree problem which on quasi-bipartite graphs has approximation ratio 1.28. The complexity of Robins and Zelikovsky’s algorithm is O(m n2), where m and n are the numbers of terminals and non-terminals in the graph, respectively. Furthermore, Gröpl et al.[6] gave a 1.217-approximation algorithm for the special case of uniformly quasi-bipartite instances.

76.1 References

[1] Rajagopalan, Sridhar; Vazirani, Vijay V. (1999), “On the bidirected cut relaxation for the metric Steiner tree problem”, Proceedings of the Tenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 742–751.

[2] Rizzi, Romeo (2003), “On Rajagopalan and Vazirani’s 3/2-approximation bound for the Iterated 1-Steiner heuristic”, Inf. Process. Lett. 86 (6): 335–338, doi:10.1016/S0020-0190(03)00210-2.

[3] Chakrabarty, Deeparnab; Devanur, Nikhil R.; Vazirani, Vijay V. (2008), “New Geometry-Inspired Relaxations and Algo- rithms for the Metric Steiner Tree Problem”, Proc. 13th IPCO, Lecture Notes in Computer Science 5035, pp. 344–358, doi:10.1007/978-3-540-68891-4_24, ISBN 978-3-540-68886-0.

[4] Gröpl, Clemens; Hougardy, Stefan; Nierhoﬀ, Till; Prömel, Hans Jürgen (2001), “Lower Bounds for Approximation Algo- rithms for the Steiner Tree Problem”, Graph-Theoretic Concepts in Computer Science : 27th International Workshop, WG 2001, Lecture Notes in Computer Science 2204, Springer-Verlag, Lecture Notes in Computer Science 2204, pp. 217–228, doi:10.1007/3-540-45477-2_20, ISBN 978-3-540-42707-0.

[5] Robins, Gabriel; Zelikovsky, Alexander (2000), “Improved Steiner tree approximation in graphs”, Proceedings of the Eleventh Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 770–779.

[6] Gröpl, Clemens; Hougardy, Stefan; Nierhoﬀ, Till; Prömel, Hans Jürgen (2002), “Steiner trees in uniformly quasi-bipartite graphs”, Information Processing Letters 83 (4): 195–200, doi:10.1016/S0020-0190(01)00335-0.

285 Chapter 77

Ramanujan graph

In spectral graph theory, a Ramanujan graph, named after Srinivasa Ramanujan, is a regular graph whose spectral gap is almost as large as possible (see extremal graph theory). Such graphs are excellent spectral expanders.

Examples of Ramanujan graphs include the clique, the biclique Kn,n , and the Petersen graph. As Murty’s survey paper notes, Ramanujan graphs “fuse diverse branches of pure mathematics, namely, number theory, representation theory, and algebraic geometry". As observed by Toshikazu Sunada, a regular graph is Ramanujan if and only if its Ihara zeta function satisﬁes an analog of the Riemann hypothesis.[1]

77.1 Deﬁnition

Let G be a connected d -regular graph with n vertices, and let λ0 ≥ λ1 ≥ ... ≥ λn−1 be the eigenvalues of the adjacency matrix of G . Because G is connected and d -regular, its eigenvalues satisfy d = λ0 > λ1 ≥ ... ≥ λn−1 ≥ −d . Whenever there exists λi with |λi| < d , deﬁne

λ(G) = max |λi|. |λi|

77.2 Extremality of Ramanujan graphs

For a ﬁxed d and large n , the d -regular, n -vertex Ramanujan graphs nearly minimize λ(G) . If G is a d -regular graph with diameter m , a theorem due to Nilli[2] states

√ √ 2 d − 1 − 1 λ ≥ 2 d − 1 − . 1 ⌊m/2⌋

| | ≥ Gd Whenever G is d -regular and connected on at least three vertices, λ1 < d , and therefore λ(G) λ1 . Let n be the set of all connected d -regular graphs G with at least n vertices. Because the minimum diameter of graphs in Gd n approaches inﬁnity for ﬁxed d and increasing n , Nilli’s theorem implies an earlier theorem of Alon and Boppana which states

√ lim inf λ(G) ≥ 2 d − 1. n→∞ ∈Gd G n

286 77.3. CONSTRUCTIONS 287

77.3 Constructions

Constructions of Ramanujan graphs are often algebraic. Lubotzky, Phillips and Sarnak show how to construct an inﬁnite family of p +1-regular Ramanujan graphs, whenever p ≡ 1 (mod 4) is a prime. Their proof uses the Ramanujan conjecture, which led to the name of Ramanujan graphs. Morgenstern extended the construction of Lubotzky, Phillips and Sarnak to all prime powers. Adam Marcus, Daniel Spielman and Nikhil Srivastava proved the existence of d -regular bipartite Ramanujan graphs for any d .

77.4 References

[1] Audrey Terras, Zeta Functions of Graphs: A Stroll through the Garden, volume 128, Cambridge Studies in Advanced Mathematics, Cambridge University Press, (2010).

[2] A Nilli, On the second eigenualue of a graph, Discrete Mathematics 91 (1991) pp. 207-210.

• Guiliana Davidoﬀ; Peter Sarnak; Alain Valette (2003). Elementary number theory, group theory and Raman- juan graphs. LMS student texts 55. Cambridge University Press. ISBN 0-521-53143-8. OCLC 50253269.

• Alexander Lubotzky; Ralph Phillips; Peter Sarnak (1988). “Ramanujan graphs”. Combinatorica 8 (3): 261– 277. doi:10.1007/BF02126799.

• Moshe Morgenstern (1994). “Existence and Explicit Constructions of q+1 Regular Ramanujan Graphs for Every Prime Power q”. J. Combinatorial Theory, Series B 62: 44–62. doi:10.1006/jctb.1994.1054.

• T. Sunada (1985). “L-functions in geometry and some applications”. Lecture Notes in Math. Lecture Notes in Mathematics 1201: 266–284. doi:10.1007/BFb0075662. ISBN 978-3-540-16770-9. • Adam Marcus; Daniel Spielman; Nikhil Srivastava (2013). “Interlacing families I: Bipartite Ramanujan graphs of all degrees”. Foundations of Computer Science (FOCS), 2013 IEEE 54th Annual Symposium.

77.5 External links

• Survey paper by M. Ram Murty Chapter 78

Reeb graph

Reeb graph of the height function on the torus.

A Reeb graph (named after Georges Reeb) is a mathematical object reflecting the evolution of the level sets of a real-valued function on a manifold.[1] Originally introduced as a tool in Morse theory,[2] Reeb graphs found a wide variety of applications in computational geometry and computer graphics, including computer aided geometric design, topology-based shape matching,[3] topological simplification and cleaning, surface segmentation and parametrization, and efficient computation of level sets. In a special case of a function on a flat space, the Reeb graph forms a polytree and is also called a contour tree.[4]

288 78.1. FORMAL DEFINITION 289

78.1 Formal deﬁnition

Given a topological space X and a continuous function f: X → R, deﬁne an equivalence relation ∼ on X where p∼q whenever p and q belong to the same connected component of a single level set f−1(c) for some real c. The Reeb graph is the quotient space X /∼ endowed with the quotient topology.

78.2 Description for Morse functions

If f is a Morse function with distinct critical values, the Reeb graph can be described more explicitly. Its nodes, or vertices, correspond to the critical level sets f−1(c). The pattern in which the arcs, or edges, meet at the nodes/vertices reﬂects the change in topology of the level set f−1(t) as t passes through the critical value c. For example, if c is a minimum or a maximum of f, a component is created or destroyed; consequently, an arc originates or terminates at the corresponding node, which has degree 1. If c is a saddle point of index 1 and two components of f−1(t) merge at t = c as t increases, the corresponding vertex of the Reeb graph has degree 3 and looks like the letter “Y"; the same reasoning applies if the index of c is dim X−1 and a component of f−1(c) splits into two.

78.3 References

[1] Harish Doraiswamy, Vijay Natarajan, Eﬃcient algorithms for computing Reeb graphs, Computational Geometry 42 (2009) 606–616

[2] G. Reeb, Sur les points singuliers d’une forme de Pfaﬀ complètement intégrable ou d’une fonction numérique, C. R. Acad. Sci. Paris 222 (1946) 847–849

[3] Tung, Tony; Schmitt, Francis (2005). “The Augmented Multiresolution Reeb Graph Approach for Content-Based Retrieval of 3D Shapes”. International Journal of Shape Modeling (IJSM) 11 (1): 91–120.

[4] Carr, Hamish; Snoeyink, Jack; Axen, Ulrike (2000), “Computing contour trees in all dimensions”, Proc. 11th ACM-SIAM Symposium on Discrete Algorithms (SODA 2000), pp. 918–926. Chapter 79

Regular graph

In graph theory, a regular graph is a graph where each vertex has the same number of neighbors; i.e. every vertex has the same degree or valency. A regular directed graph must also satisfy the stronger condition that the indegree and outdegree of each vertex are equal to each other.[1] A regular graph with vertices of degree k is called a k‑regular graph or regular graph of degree k. Regular graphs of degree at most 2 are easy to classify: A 0-regular graph consists of disconnected vertices, a 1-regular graph consists of disconnected edges, and a 2-regular graph consists of disconnected cycles and inﬁnite chains. A 3-regular graph is known as a cubic graph. A strongly regular graph is a regular graph where every adjacent pair of vertices has the same number l of neighbors in common, and every non-adjacent pair of vertices has the same number n of neighbors in common. The smallest graphs that are regular but not strongly regular are the cycle graph and the circulant graph on 6 vertices.

The complete graph Km is strongly regular for any m . A theorem by Nash-Williams says that every k‑regular graph on 2k + 1 vertices has a Hamiltonian cycle.

• 0-regular graph

• 1-regular graph

• 2-regular graph

• 3-regular graph

79.1 Existence

It is well known that the necessary and suﬃcient conditions for a k regular graph of order n to exist are that n ≥ k +1 and that nk is even. In such case it is easy to construct regular graphs by considering appropriate parameters for circulant graphs.

79.2 Algebraic properties

Let A be the adjacency matrix of a graph. Then the graph is regular if and only if j = (1,..., 1) is an eigenvector [2] of A. Its eigenvalue will be the constant degree of the graph. Eigenvectors∑ corresponding to other eigenvalues are n orthogonal to j , so for such eigenvectors v = (v1, . . . , vn) , we have i=1 vi = 0 . A regular graph of degree k is connected if and only if the eigenvalue k has multiplicity one. The “only if” direction is a consequence of the Perron–Frobenius theorem.[2] There is also a criterion for regular and connected graphs : a graph is connected and regular if and only if the matrix of ones J, with Jij = 1 , is in the adjacency algebra of the graph (meaning it is a linear combination of powers of A).[3]

290 79.3. GENERATION 291

Let G be a k-regular graph with diameter D and eigenvalues of adjacency matrix k = λ0 > λ1 ≥ · · · ≥ λn−1 . If G is not bipartite, then

− D ≤ log (n 1) + 1. [4] log(λ0/λ1)

79.3 Generation

Regular graphs may be generated by the GenReg program.[5]

79.4 See also

• Random regular graph

• Strongly regular graph • Moore graph

• Cage graph • Highly irregular graph

79.5 References

[1] Chen, Wai-Kai (1997). Graph Theory and its Engineering Applications. World Scientiﬁc. p. 29. ISBN 978-981-02-1859- 1.

[2] Cvetković, D. M.; Doob, M.; and Sachs, H. Spectra of Graphs: Theory and Applications, 3rd rev. enl. ed. New York: Wiley, 1998.

[3] Curtin, Brian (2005), “Algebraic characterizations of graph regularity conditions”, Designs, Codes and Cryptography 34 (2-3): 241–248, doi:10.1007/s10623-004-4857-4, MR 2128333.

[4] http://personal.plattsburgh.edu/quenelgt/pubpdf/diamest.pdf

[5] Meringer, Markus (1999). “Fast generation of regular graphs and construction of cages” (PDF). Journal of Graph Theory 30 (2): 137–146. doi:10.1002/(SICI)1097-0118(199902)30:2<>1.0.CO;2-G.

79.6 External links

• Weisstein, Eric W., “Regular Graph”, MathWorld.

• Weisstein, Eric W., “Strongly Regular Graph”, MathWorld. • GenReg software and data by Markus Meringer.

• Nash-Williams, Crispin (1969). “University of Waterloo Research Report”. Waterloo, Ontario: University of Waterloo. |contribution= ignored (help) Chapter 80

Scale-free network

A scale-free network is a network whose degree distribution follows a power law, at least asymptotically. That is, the fraction P(k) of nodes in the network having k connections to other nodes goes for large values of k as

P (k) ∼ k−γ where γ is a parameter whose value is typically in the range 2 < γ < 3, although occasionally it may lie outside these bounds.[1][2] Many networks have been reported to be scale-free, although statistical analysis has refuted many of these claims and seriously questioned others.[3] Preferential attachment and the ﬁtness model have been proposed as mechanisms to explain conjectured power law degree distributions in real networks.

80.1 History

In studies of the networks of citations between scientific papers, Derek de Solla Price showed in 1965 that the number of links to papers—i.e., the number of citations they receive—had a heavy-tailed distribution following a Pareto distribution or power law, and thus that the citation network is scale-free. He did not however use the term “scale-free network”, which was not coined until some decades later. In a later paper in 1976, Price also proposed a mechanism to explain the occurrence of power laws in citation networks, which he called “cumulative advantage” but which is today more commonly known under the name preferential attachment. Recent interest in scale-free networks started in 1999 with work by Albert-László Barabási and colleagues at the University of Notre Dame who mapped the topology of a portion of the World Wide Web,[4] finding that some nodes, which they called “hubs”, had many more connections than others and that the network as a whole had a power-law distribution of the number of links connecting to a node. After finding that a few other networks, including some social and biological networks, also had heavy-tailed degree distributions, Barabási and collaborators coined the term “scale-free network” to describe the class of networks that exhibit a power-law degree distribution. Amaral et al. showed that most of the real-world networks can be classified into two large categories according to the decay of degree distribution P(k) for large k. Barabási and Albert proposed a generative mechanism to explain the appearance of power-law distributions, which they called "preferential attachment" and which is essentially the same as that proposed by Price. Analytic solutions for this mechanism (also similar to the solution of Price) were presented in 2000 by Dorogovtsev, Mendes and Samukhin [5] and independently by Krapivsky, Redner, and Leyvraz, and later rigorously proved by mathematician Béla Bollobás.[6] Notably, however, this mechanism only produces a specific subset of networks in the scale-free class, and many alternative mechanisms have been discovered since.[7] The history of scale-free networks also includes some disagreement. On an empirical level, the scale-free nature of several networks has been called into question. For instance, the three brothers Faloutsos believed that the Internet had a power law degree distribution on the basis of traceroute data; however, it has been suggested that this is a layer 3 illusion created by routers, which appear as high-degree nodes while concealing the internal layer 2 structure of the ASes they interconnect. [8] On a theoretical level, refinements to the abstract definition of scale-free have been

292 80.2. CHARACTERISTICS 293

proposed. For example, Li et al. (2005) recently offered a potentially more precise “scale-free metric”. Briefly, let G be a graph with edge set E, and denote the degree of a vertex v (that is, the number of edges incident to v ) by deg(v) . Define

∑ s(G) = deg(u) · deg(v). (u,v)∈E

This is maximized when high-degree nodes are connected to other high-degree nodes. Now deﬁne

s(G) S(G) = , smax where sₐₓ is the maximum value of s(H) for H in the set of all graphs with degree distribution identical to G. This gives a metric between 0 and 1, where a graph G with small S(G) is “scale-rich”, and a graph G with S(G) close to 1 is “scale-free”. This deﬁnition captures the notion of self-similarity implied in the name “scale-free”.

80.2 Characteristics

Random network (a) and scale-free network (b). In the scale-free network, the larger hubs are highlighted.

The most notable characteristic in a scale-free network is the relative commonness of vertices with a degree that greatly exceeds the average. The highest-degree nodes are often called “hubs”, and are thought to serve specific purposes in their networks, although this depends greatly on the domain. The scale-free property strongly correlates with the network’s robustness to failure. It turns out that the major hubs are closely followed by smaller ones. These smaller hubs, in turn, are followed by other nodes with an even smaller degree and so on. This hierarchy allows for a fault tolerant behavior. If failures occur at random and the vast majority of nodes are those with small degree, the likelihood that a hub would be affected is almost negligible. Even if a hub-failure occurs, the network will generally not lose its connectedness, due to the remaining hubs. On the other hand, if we choose a few major hubs and take them out of the network, the network is turned into a set of rather isolated graphs. Thus, hubs are both a strength and a weakness of scale-free networks. These properties have been studied analytically using percolation theory by Cohen et al.[9][10] and by Callaway et al.[11] Another important characteristic of scale-free networks is the clustering coefficient distribution, which decreases as the node degree increases. This distribution also follows a power law. This implies that the low-degree nodes belong to very dense sub-graphs and those sub-graphs are connected to each other through hubs. Consider a social network in which nodes are people and links are acquaintance relationships between people. It is easy to see that people tend to form communities, i.e., small groups in which everyone knows everyone (one can think of such community as 294 CHAPTER 80. SCALE-FREE NETWORK

Complex network degree distribution of random and scale-free a complete graph). In addition, the members of a community also have a few acquaintance relationships to people outside that community. Some people, however, are connected to a large number of communities (e.g., celebrities, politicians). Those people may be considered the hubs responsible for the small-world phenomenon. At present, the more specific characteristics of scale-free networks vary with the generative mechanism used to create them. For instance, networks generated by preferential attachment typically place the high-degree vertices in the middle of the network, connecting them together to form a core, with progressively lower-degree nodes making up the regions between the core and the periphery. The random removal of even a large fraction of vertices impacts the overall connectedness of the network very little, suggesting that such topologies could be useful for security, while targeted attacks destroys the connectedness very quickly. Other scale-free networks, which place the high-degree vertices at the periphery, do not exhibit these properties. Similarly, the clustering coefficient of scale-free networks can vary significantly depending on other topological details. A final characteristic concerns the average distance between two vertices in a network. As with most disordered networks, such as the small world network model, this distance is very small relative to a highly ordered network such as a lattice graph. Notably, an uncorrelated power-law graph having 2 < γ < 3 will have ultrasmall diameter d ~ ln ln N where N is the number of nodes in the network, as proved by Cohen and Havlin. The diameter of a growing scale-free network might be considered almost constant in practice.

80.3 Examples

Although many real-world networks are thought to be scale-free, the evidence often remains inconclusive, primarily due to the developing awareness of more rigorous data analysis techniques.[3] As such, the scale-free nature of many networks is still being debated by the scientiﬁc community. A few examples of networks claimed to be scale-free include:

• Social networks, including collaboration networks. Two examples that have been studied extensively are the collaboration of movie actors in ﬁlms and the co-authorship by mathematicians of papers. • Many kinds of computer networks, including the internet and the webgraph of the World Wide Web. • Some ﬁnancial networks such as interbank payment networks [12][13] 80.4. GENERATIVE MODELS 295

• Protein-protein interaction networks.

• Semantic networks.[14]

• Airline networks.

Scale free topology has been also found in high temperature superconductors.[15] The qualities of a high-temperature superconductor — a compound in which electrons obey the laws of quantum physics, and ﬂow in perfect synchrony, without friction — appear linked to the fractal arrangements of seemingly random oxygen atoms and lattice distortion.[16]

80.4 Generative models

These scale-free networks do not arise by chance alone. Erdős and Rényi (1960) studied a model of growth for graphs in which, at each step, two nodes are chosen uniformly at random and a link is inserted between them. The properties of these random graphs are different from the properties found in scale-free networks, and therefore a model for this growth process is needed. The mostly widely known generative model for a subset of scale-free networks is Barabási and Albert’s (1999) rich get richer generative model in which each new Web page creates links to existing Web pages with a probability distribution which is not uniform, but proportional to the current in-degree of Web pages. This model was originally discovered by Derek J. de Solla Price in 1965 under the term cumulative advantage, but did not reach popularity until Barabási rediscovered the results under its current name (BA Model). According to this process, a page with many in-links will attract more in-links than a regular page. This generates a power-law but the resulting graph differs from the actual Web graph in other properties such as the presence of small tightly connected communities. More general models and networks characteristics have been proposed and studied (for a review see the book by Dorogovtsev and Mendes). A somewhat different generative model for Web links has been suggested by Pennock et al. (2002). They examined communities with interests in a specific topic such as the home pages of universities, public companies, newspapers or scientists, and discarded the major hubs of the Web. In this case, the distribution of links was no longer a power law but resembled a normal distribution. Based on these observations, the authors proposed a generative model that mixes preferential attachment with a baseline probability of gaining a link. Another generative model is the copy model studied by Kumar et al.[17] (2000), in which new nodes choose an existent node at random and copy a fraction of the links of the existent node. This also generates a power law. Interestingly, the growth of the networks (adding new nodes) is not a necessary condition for creating a scale-free network. Dangalchev[18] (2004) gives examples of generating static scale-free networks. Another possibility (Caldarelli et al. 2002) is to consider the structure as static and draw a link between vertices according to a particular property of the two vertices involved. Once specified the statistical distribution for these vertices properties (fitnesses), it turns out that in some circumstances also static networks develop scale-free properties. Kasthurirathna and Piraveenan [19] have shown that in socio-ecological systems, the drive towards improved rationality on average might be an evolutionary reason for the emergence of scale-free properties. They did this by simulating a number of strategic games on an initially random network with distributed bounded rationality, then re-wiring the network so that the network on average converged towards Nash equilibria, despite the bounded rationality of nodes. They observed that this re-wiring process results in scale-free networks.

80.5 Generalized scale-free model

There has been a burst of activity in the modeling of scale-free complex networks. The recipe of Barabási and Albert[20] has been followed by several variations and generalizations[21][22][23][24] and the revamping of previous mathematical works.[25] As long as there is a power law distribution in a model, it is a scale-free network, and a model of that network is a scale-free model. 296 CHAPTER 80. SCALE-FREE NETWORK

80.5.1 Features

Many real networks are scale-free networks, which require scale-free models to describe them. There are two ingre- dients needed to build up a scale-free model: 1. Adding or removing nodes. Usually we concentrate on growing the network, i.e. adding nodes. 2. Preferential attachment: The probability Π that new nodes will be connected to the “old” node. Note that Fitness models (see below) could work also statically, without changing the number of nodes

80.5.2 Examples

There have been several attempts to generate scale-free network properties. Here are some examples:

The Barabási–Albert model

For example, the ﬁrst scale-free model, the Barabási–Albert model, has a linear preferential attachment Π(ki) = ∑ki and adds one new node at every time step. j kj (Note, another general feature of Π(k) in real networks is that Π(0) ≠ 0 , i.e. there is a nonzero probability that a new node attaches to an isolated node. Thus in general Π(k) has the form Π(k) = A + kα , where A is the initial attractiveness of the node.)

Two-level network model

Dangalchev[18] builds a 2-L model by adding a second-order preferential attachment. The attractiveness of a node in the 2-L model depends not only on the number of nodes linked to it but also on the number of links in each of these nodes. ∑ k +C k ∑i (i,j∑) j Π(ki) = 2 , where C is a coeﬃcient between 0 and 1. j kj +C j kj

Non-linear preferential attachment

The Barabási–Albert model assumes that the probability Π(k) that a node attaches to node i is proportional to the degree k of node i . This assumption involves two hypotheses: ﬁrst, that Π(k) depends on k , in contrast to random graphs in which Π(k) = p , and second, that the functional form of Π(k) is linear in k . The precise form of Π(k) is not necessarily linear, and recent studies have demonstrated that the degree distribution depends strongly on Π(k) Krapivsky, Redner, and Leyvraz[23] demonstrate that the scale-free nature of the network is destroyed for nonlinear preferential attachment. The only case in which the topology of the network is scale free is that in which the preferential attachment is asymptotically linear, i.e. Π(ki) ∼ a∞ki as ki → ∞ . In this case the rate equation leads to

µ P (k) ∼ k−γ with γ = 1 + . a∞ This way the exponent of the degree distribution can be tuned to any value between 2 and ∞ .

Hierarchical network model

There is another kind of scale-free model, which grows according to some patterns, such as the hierarchical network model.[26] The iterative construction leading to a hierarchical network. Starting from a fully connected cluster of five nodes, we create four identical replicas connecting the peripheral nodes of each cluster to the central node of the original cluster. From this, we get a network of 25 nodes (N = 25). Repeating the same process, we can create four more replicas of the original cluster - the four peripheral nodes of each one connect to the central node of the nodes created in the first step. This gives N = 125, and the process can continue indefinitely. 80.6. SCALE-FREE IDEAL NETWORK 297

Fitness model

The idea is that the link between two vertices is assigned not randomly with a probability p equal for all the couple of vertices. Rather, for every vertex j there is an intrinsic ﬁtness x and a link between vertex i and j is created with [27] a probability p(xi, xj) . Note that the model is both In the case of World Trade Web it is possible to reconstruct all the properties by using as ﬁtnesses of the country their GDP, and taking

δxixj [28] p(xi, xj) = . 1+δxixj

Hyperbolic geometric graphs

Main article: Hyperbolic geometric graph

Assuming that a network has an underlying hyperbolic geometry, one can use the framework of spatial networks to generate scale-free degree distributions. This heterogeneous degree distribution then simply reﬂects the negative curvature and metric properties of the underlying hyperbolic geometry.[29]

80.6 Scale-free ideal network

In the context of network theory a scale-free ideal network is a random network with a degree distribution following the scale-free ideal gas density distribution. These networks are able to reproduce city-size distributions and electoral results by unraveling the size distribution of social groups with information theory on complex networks when a competitive cluster growth process is applied to the network.[30][31] In models of scale-free ideal networks it is possible to demonstrate that Dunbar’s number is the cause of the phenomenon known as the 'six degrees of separation'.

80.7 See also

• Random graph • Erdős–Rényi model • Non-Linear Preferential Attachment • Bose-Einstein condensation: a network theory approach • Scale invariance • Complex network • Webgraph • Barabási–Albert model • Bianconi–Barabási model

80.8 Notes

[1] Onnela, J. -P.; Saramaki, J.; Hyvonen, J.; Szabo, G.; Lazer, D.; Kaski, K.; Kertesz, J.; Barabasi, A. -L. (2007). “Structure and tie strengths in mobile communication networks”. Proceedings of the National Academy of Sciences 104 (18): 7332– 7336. arXiv:physics/0610104. Bibcode:2007PNAS..104.7332O. doi:10.1073/pnas.0610245104. PMC 1863470. PMID 17456605.

[2] Choromański, K.; Matuszak, M.; MiȩKisz, J. (2013). “Scale-Free Graph with Preferential Attachment and Evolving Inter- nal Vertex Structure”. Journal of Statistical Physics 151 (6): 1175. Bibcode:2013JSP...151.1175C. doi:10.1007/s10955- 013-0749-1.

[3] Clauset, Aaron; Cosma Rohilla Shalizi; M. E. J Newman (2007-06-07). “Power-law distributions in empirical data”. 0706.1062. arXiv:0706.1062. Bibcode:2009SIAMR..51..661C. doi:10.1137/070710111. 298 CHAPTER 80. SCALE-FREE NETWORK

[4] Barabási, Albert-László; Albert, Réka. (October 15, 1999). “Emergence of scaling in random networks”. Science 286 (5439): 509–512. arXiv:cond-mat/9910332. Bibcode:1999Sci...286..509B. doi:10.1126/science.286.5439.509. MR 2091634. PMID 10521342.

[5] Dorogovtsev, S.; Mendes, J.; Samukhin, A. (2000). “Structure of Growing Networks with Preferential Linking”. Physical Review Letters 85 (21): 4633–4636. arXiv:cond-mat/0004434. Bibcode:2000PhRvL..85.4633D. doi:10.1103/PhysRevLett.85.4633. PMID 11082614.

[6] Bollobás, B.; Riordan, O.; Spencer, J.; Tusn�Dy, G. (2001). “The degree sequence of a scale-free random graph process”. Random Structures and Algorithms 18 (3): 279–290. doi:10.1002/rsa.1009. MR 1824277. replacement character in |last4= at position 5 (help)

[7] Dorogovtsev, S. N.; Mendes, J. F. F. (2002). “Evolution of networks”. Advances in Physics 51 (4): 1079. doi:10.1080/00018730110112519.

[8] Willinger, Walter; David Alderson; John C. Doyle (May 2009). “Mathematics and the Internet: A Source of Enormous Confusion and Great Potential” (PDF). Notices of the AMS (American Mathematical Society) 56 (5): 586–599. Retrieved 2011-02-03.

[9] Cohen, Reoven; Erez, K.; ben-Avraham, D.; Havlin, S. (2000). “Resilience of the Internet to Random Breakdowns”. Phys. Rev. Lett. 85: 4626–8. arXiv:cond-mat/0007048. Bibcode:2000PhRvL..85.4626C. doi:10.1103/PhysRevLett.85.4626.

[10] Cohen, Reoven; Erez, K.; ben-Avraham, D.; Havlin, S. (2001). “Breakdown of the Internet under Intentional Attack”. Phys. Rev. Lett. 86: 3682–5. arXiv:cond-mat/0010251. Bibcode:2001PhRvL..86.3682C. doi:10.1103/PhysRevLett.86.3682. PMID 11328053.

[11] Callaway, Duncan S.; Newman, M. E. J.; Strogatz, S. H.; Watts, D. J. (2000). “Network Robustness and Fragility: Perco- lation on Random Graphs”. Phys. Rev. Lett. 85: 5468–71. arXiv:cond-mat/0007300. Bibcode:2000PhRvL..85.5468C. doi:10.1103/PhysRevLett.85.5468.

[12] De Masi, Giulia; et. al (2006). “Fitness model for the Italian interbank money market”. Physical Review E 74: 066112. doi:10.1103/PhysRevE.74.066112.

[13] Soramäki, Kimmo; et. al (2007). “The topology of interbank payment ﬂows”. Physica A: Statistical Mechanics and its Applications 379 (1): 317–333. Bibcode:2007PhyA..379..317S. doi:10.1016/j.physa.2006.11.093.

[14] Steyvers, Mark; Joshua B. Tenenbaum (2005). “The Large-Scale Structure of Semantic Networks: Statistical Analyses and a Model of Semantic Growth”. Cognitive Science 29 (1): 41–78. doi:10.1207/s15516709cog2901_3.

[15] Fratini, Michela, Poccia, Nicola, Ricci, Alessandro, Campi, Gaetano, Burghammer, Manfred, Aeppli, Gabriel Bianconi, Antonio (2010). “Scale-free structural organization of oxygen interstitials in La2CuO4+y”. Nature 466 (7308): 841–4. arXiv:1008.2015. Bibcode:2010Natur.466..841F. doi:10.1038/nature09260. PMID 20703301.

[16] Poccia, Nicola, Ricci, Alessandro, Campi, Gaetano, Fratini, Michela, Puri, Alessandro, Di Gioacchino, Daniele, Mar- celli, Augusto, Reynolds, Michael, Burghammer, Manfred, Saini, Naurang L., Aeppli, Gabriel Bianconi, Antonio, (2012). “Optimum inhomogeneity of local lattice distortions in La2CuO4+y”. Proc. Natl. Acad. Sci. U.S.A. 109 (39): 15685– 15690. arXiv:1208.0101. doi:10.1073/pnas.1208492109.

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[18] Dangalchev Ch., Generation models for scale-free networks, Physica A 338, 659 (2004).

[19] Kasthurirathna, Dharshana; Piraveenan, Mahendra. (2015). “Emergence of scale-free characteristics in socioecological systems with bounded rationality”. Scientiﬁc Reports. In Press.

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[21] R. Albert, and A.L. Barabási, Phys. Rev. Lett. 85, 5234(2000).

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[24] B. Tadic, Physica A 293, 273(2001).

[25] S. Bomholdt and H. Ebel, cond-mat/0008465; H.A. Simon, Bimetrika 42, 425(1955).

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• Manev R., Manev H. (2005). “The meaning of mammalian adult neurogenesis and the function of newly added neurons: the “small-world” network”. Med. Hypotheses 64 (1): 114–7. doi:10.1016/j.mehy.2004.05.013. PMID 15533625. • Matlis, Jan (November 4, 2002). “Scale-Free Networks”.

• Newman, Mark E.J. (2003). “The structure and function of complex networks”. arXiv:cond-mat/0303516 [cond-mat.stat-mech]. • Pastor-Satorras, R., Vespignani, A. (2004). Evolution and Structure of the Internet: A Statistical Physics Ap- proach. Cambridge University Press. ISBN 0-521-82698-5. • Pennock, D.M., Flake, G.W., Lawrence, S., Glover, E.J., Giles, C.L. (2002). “Winners don't take all: Charac- terizing the competition for links on the web”. Proc. Natl. Acad. Sci. U.S.A. 99 (8): 5207–11. Bibcode:2002PNAS...99.5207P. doi:10.1073/pnas.032085699. PMC 122747. PMID 16578867.

• Robb, John. Scale-Free Networks and Terrorism, 2004. • Keller, E.F. (2005). “Revisiting “scale-free” networks”. BioEssays 27 (10): 1060–8. doi:10.1002/bies.20294. PMID 16163729. • Onody, R.N., de Castro, P.A. (2004). “Complex Network Study of Brazilian Soccer Player”. Phys. Rev. E 70: 037103. arXiv:cond-mat/0409609. Bibcode:2004PhRvE..70c7103O. doi:10.1103/PhysRevE.70.037103. • Reuven Cohen, Shlomo Havlin (2003). “Scale-Free Networks are Ultrasmall”. Phys. Rev. Lett. 90 (5): 058701. arXiv:cond-mat/0205476. Bibcode:2003PhRvL..90e8701C. doi:10.1103/PhysRevLett.90.058701. PMID 12633404.

• Kasthurirathna, D., Piraveenan, M. (2015). “Complex Network Study of Brazilian Soccer Player”. Sci. Rep. In Press.

80.10 External links

• snGraph Optimal software to manage scale-free networks.

• The Erdős Webgraph Server describing the hyperlink structure of a weekly updated, constantly increasing portion of the WWW. Chapter 81

Self-complementary graph

A self-complementary graph: the blue N is isomorphic to its complement, the dashed red Z.

A self-complementary graph is a graph which is isomorphic to its complement. The simplest non-trivial self- complementary graphs are the 4-vertex path graph and the 5-vertex cycle graph.

301 302 CHAPTER 81. SELF-COMPLEMENTARY GRAPH

81.1 Examples

Every Paley graph is self-complementary.[1] For example, the 3 × 3 rook’s graph (the Paley graph of order nine) is self-complementary, by a symmetry that keeps the center vertex in place but exchanges the roles of the four side midpoints and four corners of the grid.[2] All strongly regular self-complementary graphs with fewer than 37 vertices are Paley graphs; however, there are strongly regular graphs on 37, 41, and 49 vertices that are not Paley graphs.[3] The Rado graph is an inﬁnite self-complementary graph.

81.2 Properties

An n-vertex self-complementary graph has exactly half number of edges of the complete graph, i.e., n(n − 1)/4 edges, and (if there is more than one vertex) it must have diameter either 2 or 3.[1] Since n(n −1) must be divisible by 4, n must be congruent to 0 or 1 mod 4; for instance, a 6-vertex graph cannot be self-complementary.

81.3 Computational complexity

The problems of checking whether two self-complementary graphs are isomorphic and of checking whether a given graph is self-complementary are polynomial-time equivalent to the general graph isomorphism problem.[4]

81.4 References

[1] Sachs, Horst (1962), "Über selbstkomplementäre Graphen”, Publicationes Mathematicae Debrecen 9: 270–288, MR 0151953.

[2] Shpectorov, S. (1998), “Complementary l1-graphs”, Discrete Mathematics 192 (1-3): 323–331, doi:10.1016/S0012-365X(98)0007X- 1, MR 1656740.

[3] Rosenberg, I. G. (1982), “Regular and strongly regular selfcomplementary graphs”, Theory and practice of combinatorics, North-Holland Math. Stud. 60, Amsterdam: North-Holland, pp. 223–238, MR 806985.

[4] Colbourn, Marlene J.; Colbourn, Charles J. (1978), “Graph isomorphism and self-complementary graphs”, SIGACT News 10 (1): 25–29, doi:10.1145/1008605.1008608.

81.5 External links

• Weisstein, Eric W., “Self-Complementary Graph”, MathWorld. Chapter 82

Semi-symmetric graph

The Folkman graph, the smallest semi-symmetric graph.

In the mathematical ﬁeld of graph theory, a semi-symmetric graph is an undirected graph that is edge-transitive and regular, but not vertex-transitive. In other words, a graph is semi-symmetric if each vertex has the same number of incident edges, and there is a symmetry taking any of its edges to any other of its edges, but there is some pair of vertices such that no symmetry maps the ﬁrst into the second.

303 304 CHAPTER 82. SEMI-SYMMETRIC GRAPH

82.1 Properties

A semi-symmetric graph must be bipartite, and its automorphism group must act transitively on each of the two vertex sets of the bipartition. For instance, in the diagram of the Folkman graph shown here, green vertices can not be mapped to red ones by any automorphism, but every two vertices of the same color are symmetric with each other.

82.2 History

Semi-symmetric graphs were ﬁrst studied E. Dauber, a student of F. Harary, in a paper, no longer available, titled “On line- but not point-symmetric graphs”. This was seen by Jon Folkman, whose paper, published in 1967, includes the smallest semi-symmetric graph, now known as the Folkman graph, on 20 vertices.[1] The term “semi-symmetric” was ﬁrst used by Klin et al. in a paper they published in 1978.[2]

82.3 Cubic graphs

The smallest cubic semi-symmetric graph (that is, one in which each vertex is incident to exactly three edges) is the Gray graph on 54 vertices. It was ﬁrst observed to be semi-symmetric by Bouwer (1968). It was proven to be the smallest cubic semi-symmetric graph by Dragan Marušič and Aleksander Malnič.[3] All the cubic semi-symmetric graphs on up to 768 vertices are known. According to Conder, Malnič, Marušič and Potočnik, the four smallest possible cubic semi-symmetric graphs after the Gray graph are the Ioﬁnova–Ivanov graph on 110 vertices, the Ljubljana graph on 112 vertices,[4] a graph on 120 vertices with girth 8 and the Tutte 12-cage.[5]

82.4 References

[1] Folkman, J. (1967), “Regular line-symmetric graphs”, Journal of Combinatorial Theory 3 (3): 215–232, doi:10.1016/S0021- 9800(67)80069-3.

[2] Klin, Lauri & Ziv-Av (2011). “Links between two semisymmetric graphs on 112 vertices through the lens of association schemes” (PDF). Retrieved 17 August 2015.

[3] Bouwer, I. Z. (1968), “An edge but not vertex transitive cubic graph”, Bulletin of the Canadian Mathematical Society 11: 533–535, doi:10.4153/CMB-1968-063-0.

[4] Conder, M.; Malnič, A.; Marušič, D.; Pisanski, T.; Potočnik, P. (2002), “The Ljubljana Graph” (PDF), IMFM Preprints (Ljubljana: Institute of Mathematics, Physics and Mechanics) 40 (845).

[5] Conder, Marston; Malnič, Aleksander; Marušič, Dragan; Potočnik, Primož (2006), “A census of semisymmetric cubic graphs on up to 768 vertices”, Journal of Algebraic Combinatorics 23 (3): 255–294, doi:10.1007/s10801-006-7397-3.

82.5 External links

• Weisstein, Eric W., “Semisymmetric Graph”, MathWorld. Chapter 83

Series-parallel graph

In graph theory, series-parallel graphs are graphs with two distinguished vertices called terminals, formed recursively by two simple composition operations. They can be used to model series and parallel electric circuits.

83.1 Deﬁnition and terminology

In this context, the term graph means multigraph. There are several ways to define series-parallel graphs. The following definition basically follows the one used by David Eppstein.[1] A two-terminal graph (TTG) is a graph with two distinguished vertices, s and t called source and sink, respectively. The parallel composition Pc = Pc(X,Y) of two TTGs X and Y is a TTG created from the disjoint union of graphs X and Y by merging the sources of X and Y to create the source of Pc and merging the sinks of X and Y to create the sink of Pc. The series composition Sc = Sc(X,Y) of two TTGs X and Y is a TTG created from the disjoint union of graphs X and Y by merging the sink of X with the source of Y. The source of X becomes the source of Sc and the sink of Y becomes the sink of Sc. A two-terminal series-parallel graph (TTSPG) is a graph that may be constructed by a sequence of series and parallel compositions starting from a set of copies of a single-edge graph K2 with assigned terminals. Definition 1. Finally, a graph is called series-parallel (sp-graph), if it is a TTSPG when some two of its vertices are regarded as source and sink. In a similar way one may define series-parallel digraphs, constructed from copies of single-arc graphs, with arcs directed from the source to the sink.

83.1.1 Alternative deﬁnition

The following deﬁnition speciﬁes the same class of graphs.[2]

Deﬁnition 2. A graph is an sp-graph, if it may be turned into K2 by a sequence of the following operations:

• Replacement of a pair of parallel edges with a single edge that connects their common endpoints

• Replacement of a pair of edges incident to a vertex of degree 2 other than s or t with a single edge.

83.2 Properties

Every series-parallel graph has treewidth at most 2 and branchwidth at most 2.[3] Indeed, a graph has treewidth at most 2 if and only if it has branchwidth at most 2, if and only if every biconnected component is a series-parallel

305 306 CHAPTER 83. SERIES-PARALLEL GRAPH s s series + composition s t t

parallel composition s

t t

Series and parallel composition operations for series-parallel graphs. graph.[4][5] The maximal series-parallel graphs, graphs to which no additional edges can be added without destroying their series-parallel structure, are exactly the 2-trees. 83.3. RESEARCH INVOLVING SERIES-PARALLEL GRAPHS 307

[3] Series-parallel graphs are characterised by having no subgraph homeomorphic to K4. Series parallel graphs may also be characterized by their ear decompositions.[1]

83.3 Research involving series-parallel graphs

SPGs may be recognized in linear time[6] and their series-parallel decomposition may be constructed in linear time as well. Besides being a model of certain types of electric networks, these graphs are of interest in computational complexity theory, because a number of standard graph problems are solvable in linear time on SPGs,[7] including ﬁnding of the maximum matching, maximum independent set, minimum dominating set and Hamiltonian completion. Some of these problems are NP-complete for general graphs. The solution capitalizes on the fact that if the answers for one of these problems are known for two SP-graphs, then one can quickly ﬁnd the answer for their series and parallel compositions. The series-parallel networks problem refers to a graph enumeration problem which asks for the number of series- parallel graphs that can be formed using a given number of edges.

83.4 Generalization

The generalized series-parallel graphs (GSP-graphs) are an extension of the SPGs[8] with the same algorithmic efficiency for the mentioned problems. The class of GSP-graphs include the classes of SP-graphs and outerplanar graphs. GSP graphs may be specified by the Definition 2 augmented with the third operation of deletion of a dangling vertex (vertex of degree 1). Alternatively, Definition 1 may be augmented with the following operation.

• The source merge S = M(X,Y) of two TTGs X and Y is a TTG created from the disjoint union of graphs X and Y by merging the source of X with the source of Y. The source and sink of X become the source and sink of P respectively.

An SPQR tree is a tree structure that can be deﬁned for an arbitrary 2-vertex-connected graph. It has S nodes that are analogous to the series composition operations in series-parallel graphs, P nodes that are analogous to the parallel composition operations in series-parallel graphs, and R nodes that do not correspond to series-parallel composition operations. A 2-connected graph is series-parallel if and only if there are no R nodes in its SPQR tree.

83.5 See also

• Threshold graph

• Cograph

• Hanner polytope

• Series-parallel partial order

83.6 References

[1] Eppstein, David (1992). “Parallel recognition of series-parallel graphs” (PDF). Information and Computation 98 (1): 41– 55. doi:10.1016/0890-5401(92)90041-D.

[2] Duﬃn, R. J. (1965). “Topology of Series-Parallel Networks”. Journal of Mathematical Analysis and Applications 10 (2): 303–313. doi:10.1016/0022-247X(65)90125-3. 308 CHAPTER 83. SERIES-PARALLEL GRAPH

[3] Brandstädt, Andreas; Le, Van Bang; Spinrad, Jeremy P. (1999). Graph classes: a survey. SIAM Monographs on Discrete Mathematics. and Applications 3. Philadelphia, PA: Society for Industrial and Applied Mathematics. pp. 172–174. ISBN 978-0-898714-32-6. Zbl 0919.05001.

[4] Bodlaender, H. (1998). “A partial k-arboretum of graphs with bounded treewidth”. Theoretical Computer Science 209 (1–2): 1–45. doi:10.1016/S0304-3975(97)00228-4.

[5] Hall, Rhiannon; Oxley, James; Semple, Charles; Whittle, Geoﬀ (2002). “On matroids of branch-width three”. Journal of Combinatorial Theory, Series B 86 (1): 148–171. doi:10.1006/jctb.2002.2120.

[6] Valdes, Jacobo; Tarjan, Robert E.; Lawler, Eugene L. (1982). “The recognition of series parallel digraphs”. SIAM Journal on Computing 11 (2): 289–313. doi:10.1137/0211023.

[7] Takamizawa, K.; Nishizeki, T.; Saito, N. (1982). “Linear-time computability of combinatorial problems on series-parallel graphs”. Journal of the ACM 29 (3): 623–641. doi:10.1145/322326.322328.

[8] Korneyenko, N. M. (1994). “Combinatorial algorithms on a class of graphs”. Discrete Applied Mathematics 54 (2–3): 215– 217. doi:10.1016/0166-218X(94)90022-1. Translated from Notices of the BSSR Academy of Sciences, Ser. Phys.-Math. Sci., (1984) no. 3, pp. 109-111 (Russian) Chapter 84

Simplex graph

A graph G and the corresponding simplex graph κ(G). The blue-colored node in κ(G) corresponds to the zero-vertex clique in G (the empty set), and the magenta node corresponds to the 3-vertex clique.

In graph theory, a branch of mathematics, the simplex graph κ(G) of an undirected graph G is itself a graph, with one node for each clique (a set of mutually adjacent vertices) in G. Two nodes of κ(G) are linked by an edge whenever the corresponding two cliques diﬀer in the presence or absence of a single vertex. The empty set is included as one of the cliques of G that are used to form the clique graph, as is every set of one vertex and every set of two adjacent vertices. Therefore, the simplex graph contains within it a subdivision of G itself. The simplex graph of a complete graph is a hypercube graph, and the simplex graph of a cycle graph of length four or more is a gear graph. The simplex graph of the complement graph of a path graph is a Fibonacci cube. The complete subgraphs of G can be given the structure of a median algebra: the median of three cliques A, B, and C is formed by the vertices that belong to a majority of the three cliques.[1] Any two vertices belonging to this median set must both belong to at least one of A, B, or C, and therefore must be linked by an edge, so the median of three cliques is itself a clique. The simplex graph is the median graph corresponding to this median algebra structure. When G is the complement graph of a bipartite graph, the cliques of G can be given a stronger structure as a distributive lattice,[2] and in this case the simplex graph is the graph of the lattice. As is true for median graphs more generally, every simplex graph is itself bipartite. The simplex graph has one vertex for every simplex in the clique complex X(G) of G, and two vertices are linked by an edge when one of the two corresponding simplexes is a facet of the other. Thus, the objects (vertices in the simplex

309 310 CHAPTER 84. SIMPLEX GRAPH graph, simplexes in X(G)) and relations between objects (edges in the simplex graph, inclusion relations between simplexes in X(G)) are in one-to-one correspondence between X(G) and κ(G). Simplex graphs were introduced by Bandelt & van de Vel (1989),[3] who observed that a simplex graph has no cubes if and only if the underlying graph is triangle-free, and showed that the chromatic number of the underlying graph equals the minimum number n such that the simplex graph can be isometrically embedded into a Cartesian product of n trees. As a consequence of the existence of triangle-free graphs with high chromatic number, they showed that there exist two-dimensional topological median algebras that cannot be embedded into products of ﬁnitely many real trees. Imrich, Klavžar & Mulder (1999) also use simplex graphs as part of their proof that testing whether a graph is triangle-free or whether it is a median graph may be performed equally quickly.

84.1 Notes

[1] Barthélemy, Leclerc & Monjardet (1986), page 200.

[2] Propp (1997).

[3] Imrich, Klavžar & Mulder (1999) credit the introduction of simplex graphs to a later paper, also by Bandelt and van de Vel, but this appears to be a mistake.

84.2 References

• Bandelt, H.-J.; Chepoi, V. (2008), “Metric graph theory and geometry: a survey”, in Goodman, J.E.; Pach, J.; Pollack, R., Surveys on Discrete and Computational Geometry: Twenty Years Later, Contemp. Math. 453, Providence, RI: AMS, pp. 49–86.

• Bandelt, H.-J.; van de Vel, M. (1989), “Embedding topological median algebras in products of dendrons”, Proceedings of the London Mathematical Society, s3-58 (3): 439–453, doi:10.1112/plms/s3-58.3.439.

• Barthélemy, J.-P.; Leclerc, B.; Monjardet, B. (1986), “On the use of ordered sets in problems of comparison and consensus of classiﬁcations”, Journal of Classiﬁcation 3 (2): 187–224, doi:10.1007/BF01894188.

• Imrich, Wilfried; Klavžar, Sandi; Mulder, Henry Martyn (1999), “Median graphs and triangle-free graphs”, SIAM Journal on Discrete Mathematics 12 (1): 111–118, doi:10.1137/S0895480197323494, MR 1666073.

• Propp, James (1997), “Generating random elements of ﬁnite distributive lattices”, Electronic Journal of Com- binatorics 4 (2): R15, arXiv:math.CO/9801066. Chapter 85

Skew-symmetric graph

In graph theory, a branch of mathematics, a skew-symmetric graph is a directed graph that is isomorphic to its own transpose graph, the graph formed by reversing all of its edges, under an isomorphism that is an involution without any fixed points. Skew-symmetric graphs are identical to the double covering graphs of bidirected graphs. Skew-symmetric graphs were first introduced under the name of antisymmetrical digraphs by Tutte (1967), later as the double covering graphs of polar graphs by Zelinka (1976b), and still later as the double covering graphs of bidirected graphs by Zaslavsky (1991). They arise in modeling the search for alternating paths and alternating cycles in algorithms for finding matchings in graphs, in testing whether a still life pattern in Conway’s Game of Life may be partitioned into simpler components, in graph drawing, and in the implication graphs used to efficiently solve the 2-satisfiability problem.

85.1 Deﬁnition

As deﬁned, e.g., by Goldberg & Karzanov (1996), a skew-symmetric graph G is a directed graph, together with a function σ mapping vertices of G to other vertices of G, satisfying the following properties:

1. For every vertex v, σ(v) ≠ v,

2. For every vertex v, σ(σ(v)) = v,

3. For every edge (u,v), (σ(v),σ(u)) must also be an edge.

One may use the third property to extend σ to an orientation-reversing function on the edges of G. The transpose graph of G is the graph formed by reversing every edge of G, and σ deﬁnes a graph isomorphism from G to its transpose. However, in a skew-symmetric graph, it is additionally required that the isomorphism pair each vertex with a diﬀerent vertex, rather than allowing a vertex to be mapped to itself by the isomorphism or to group more than two vertices in a cycle of isomorphism. A path or cycle in a skew-symmetric graph is said to be regular if, for each vertex v of the path or cycle, the corresponding vertex σ(v) is not part of the path or cycle.

85.2 Examples

Every directed path graph with an even number of vertices is skew-symmetric, via a symmetry that swaps the two ends of the path. However, path graphs with an odd number of vertices are not skew-symmetric, because the orientation- reversing symmetry of these graphs maps the center vertex of the path to itself, something that is not allowed for skew-symmetric graphs. Similarly, a directed cycle graph is skew-symmetric if and only if it has an even number of vertices. In this case, the number of diﬀerent mappings σ that realize the skew symmetry of the graph equals half the length of the cycle.

311 312 CHAPTER 85. SKEW-SYMMETRIC GRAPH

85.3 Polar/switch graphs, double covering graphs, and bidirected graphs

A skew-symmetric graph may equivalently be defined as the double covering graph of a polar graph (introduced by Zelinka (1974), Zelinka (1976), called a switch graph by Cook (2003)), which is an undirected graph in which the edges incident to each vertex are partitioned into two subsets. Each vertex of the polar graph corresponds to two vertices of the skew-symmetric graph, and each edge of the polar graph corresponds to two edges of the skew- symmetric graph. This equivalence is the one used by Goldberg & Karzanov (1996) to model problems of matching in terms of skew-symmetric graphs; in that application, the two subsets of edges at each vertex are the unmatched edges and the matched edges. Zelinka (following F. Zitek) and Cook visualize the vertices of a polar graph as points where multiple tracks of a train track come together: if a train enters a switch via a track that comes in from one direction, it must exit via a track in the other direction. The problem of finding non-self-intersecting smooth curves between given points in a train track comes up in testing whether certain kinds of graph drawings are valid (Hui, Schaefer & Štefankovič 2004) and may be modeled as the search for a regular path in a skew-symmetric graph. A closely related concept is the bidirected graph of Edmonds & Johnson (1970) (“polarized graph” in the terminology of Zelinka (1974), Zelinka (1976)), a graph in which each of the two ends of each edge may be either a head or a tail, independently of the other end. A bidirected graph may be interpreted as a polar graph by letting the partition of edges at each vertex be determined by the partition of endpoints at that vertex into heads and tails; however, swapping the roles of heads and tails at a single vertex (“switching” the vertex, in the terminology of Zaslavsky (1991)) produces a different bidirected graph but the same polar graph. For the correspondence between bidirected graphs and skew-symmetric graphs (i.e., their double covering graphs) see Zaslavsky (1991), Section 5, or Babenko (2006). To form the double covering graph (i.e., the corresponding skew-symmetric graph) from a polar graph G, create for each vertex v of G two vertices v0 and v1, and let σ(vi) = v₁ ₋ i. For each edge e = (u,v) of G, create two directed edges in the covering graph, one oriented from u to v and one oriented from v to u. If e is in the first subset of edges at v, these two edges are from u0 into v0 and from v1 into u1, while if e is in the second subset, the edges are from u0 into v1 and from v0 into u1. In the other direction, given a skew-symmetric graph G, one may form a polar graph that has one vertex for every corresponding pair of vertices in G and one undirected edge for every corresponding pair of edges in G. The undirected edges at each vertex of the polar graph may be partitioned into two subsets according to which vertex of the polar graph they go out of and come in to. A regular path or cycle of a skew-symmetric graph corresponds to a path or cycle in the polar graph that uses at most one edge from each subset of edges at each of its vertices.

85.4 Matching

In constructing matchings in undirected graphs, it is important to find alternating paths, paths of vertices that start and end at unmatched vertices, in which the edges at odd positions in the path are not part of a given partial matching and in which the edges at even positions in the path are part of the matching. By removing the matched edges of such a path from a matching, and adding the unmatched edges, one can increase the size of the matching. Similarly, cycles that alternate between matched and unmatched edges are of importance in weighted matching problems. As Goldberg & Karzanov (1996) showed, an alternating path or cycle in an undirected graph may be modeled as a regular path or cycle in a skew-symmetric directed graph. To create a skew-symmetric graph from an undirected graph G with a specified matching M, view G as a switch graph in which the edges at each vertex are partitioned into matched and unmatched edges; an alternating path in G is then a regular path in this switch graph and an alternating cycle in G is a regular cycle in the switch graph. Goldberg & Karzanov (1996) generalized alternating path algorithms to show that the existence of a regular path between any two vertices of a skew-symmetric graph may be tested in linear time. Given additionally a non-negative length function on the edges of the graph that assigns the same length to any edge e and to σ(e), the shortest regular path connecting a given pair of nodes in a skew-symmetric graph with m edges and n vertices may be tested in time O(m log n). If the length function is allowed to have negative lengths, the existence of a negative regular cycle may be tested in polynomial time. Along with the path problems arising in matchings, skew-symmetric generalizations of the max-flow min-cut theorem have also been studied (Goldberg & Karzanov 2004; Tutte 1967). 85.5. STILL LIFE THEORY 313

85.5 Still life theory

Cook (2003) shows that a still life pattern in Conway’s Game of Life may be partitioned into two smaller still lifes if and only if an associated switch graph contains a regular cycle. As he shows, for switch graphs with at most three edges per vertex, this may be tested in polynomial time by repeatedly removing bridges (edges the removal of which disconnects the graph) and vertices at which all edges belong to a single partition until no more such simpliﬁcations may be performed. If the result is an empty graph, there is no regular cycle; otherwise, a regular cycle may be found in any remaining bridgeless component. The repeated search for bridges in this algorithm may be performed eﬃciently using a dynamic graph algorithm of Thorup (2000). Similar bridge-removal techniques in the context of matching were previously considered by Gabow, Kaplan & Tarjan (1999).

85.6 Satisﬁability

~x2 x0

x6 ~x4 x3

~x5 ~x1 x1 x5

~x3 x4 ~x6

~x0 x2

An implication graph. Its skew symmetry can be realized by rotating the graph through a 180 degree angle and reversing all edges.

An instance of the 2-satisfiability problem, that is, a Boolean expression in conjunctive normal form with two variables or negations of variables per clause, may be transformed into an implication graph by replacing each clause u∨v by the two implications (¬u)⇒v and (¬v)⇒u . This graph has a vertex for each variable or negated variable, and a directed edge for each implication; it is, by construction, skew-symmetric, with a correspondence σ that maps each variable to its negation. As Aspvall, Plass & Tarjan (1979) showed, a satisfying assignment to the 2-satisfiability instance is equivalent to a partition of this implication graph into two subsets of vertices, S and σ(S), such that no edge starts in S and ends in σ(S). If such a partition exists, a satisfying assignment may be formed by assigning a true value to every variable in S and a false value to every variable in σ(S). This may be done if and only if no strongly connected component of the graph contains both some vertex v and its complementary vertex σ(v). If two vertices belong to the 314 CHAPTER 85. SKEW-SYMMETRIC GRAPH same strongly connected component, the corresponding variables or negated variables are constrained to equal each other in any satisfying assignment of the 2-satisfiability instance. The total time for testing strong connectivity and finding a partition of the implication graph is linear in the size of the given 2-CNF expression.

85.7 Recognition

It is NP-complete to determine whether a given directed graph is skew-symmetric, by a result of Lalonde (1981) that it is NP-complete to find a color-reversing involution in a bipartite graph. Such an involution exists if and only if the directed graph given by orienting each edge from one color class to the other is skew-symmetric, so testing skew-symmetry of this directed graph is hard. This complexity does not affect path-finding algorithms for skew- symmetric graphs, because these algorithms assume that the skew-symmetric structure is given as part of the input to the algorithm rather than requiring it to be inferred from the graph alone.

85.8 References

• Aspvall, Bengt; Plass, Michael F.; Tarjan, Robert E. (1979), “A linear-time algorithm for testing the truth of certain quantified boolean formulas”, Information Processing Letters 8 (3): 121–123, doi:10.1016/0020- 0190(79)90002-4. • Babenko, Maxim A. (2006), “Acyclic bidirected and skew-symmetric graphs: algorithms and structure”, Com- puter Science – Theory and Applications, Lecture Notes in Computer Science 3967, Springer-Verlag, pp. 23– 34, doi:10.1007/11753728_6, ISBN 978-3-540-34166-6. • Biggs, Norman (1974), Algebraic Graph Theory, London: Cambridge University Press. • Cook, Matthew (2003), “Still life theory”, New Constructions in Cellular Automata, Santa Fe Institute Studies in the Sciences of Complexity, Oxford University Press, pp. 93–118. • Edmonds, Jack; Johnson, Ellis L. (1970), “Matching: a well-solved class of linear programs”, Combinatorial Structures and their Applications: Proceedings of the Calgary Symposium, June 1969, New York: Gordon and Breach. Reprinted in Combinatorial Optimization — Eureka, You Shrink!, Springer-Verlag, Lecture Notes in Computer Science 2570, 2003, pp. 27–30, doi:10.1007/3-540-36478-1_3. • Gabow, Harold N.; Kaplan, Haim; Tarjan, Robert E. (1999), “Unique maximum matching algorithms”, Proc. 31st ACM Symp. Theory of Computing (STOC), pp. 70–78, doi:10.1145/301250.301273, ISBN 1-58113-067- 8. • Goldberg, Andrew V.; Karzanov, Alexander V. (1996), “Path problems in skew-symmetric graphs”, Combi- natorica 16 (3): 353–382, doi:10.1007/BF01261321. • Goldberg, Andrew V.; Karzanov, Alexander V. (2004), “Maximum skew-symmetric flows and matchings”, Mathematical programming 100 (3): 537–568, doi:10.1007/s10107-004-0505-z. • Hui, Peter; Schaefer, Marcus; Štefankovič, Daniel (2004), “Train tracks and confluent drawings”, Proc. 12th Int. Symp. Graph Drawing, Lecture Notes in Computer Science 3383, Springer-Verlag, pp. 318–328. • Lalonde, François (1981), “Le problème d'étoiles pour graphes est NP-complet”, Discrete Mathematics 33 (3): 271–280, doi:10.1016/0012-365X(81)90271-5, MR 602044. • Thorup, Mikkel (2000), “Near-optimal fully-dynamic graph connectivity”, Proc. 32nd ACM Symposium on Theory of Computing, pp. 343–350, doi:10.1145/335305.335345, ISBN 1-58113-184-4. • Tutte, W. T. (1967), “Antisymmetrical digraphs”, Canadian Journal of Mathematics 19: 1101–1117, doi:10.4153/CJM- 1967-101-8. • Zaslavsky, Thomas (1982), “Signed graphs”, Discrete Applied Mathematics 4: 47–74, doi:10.1016/0166-218X(82)90033- 6. • Zaslavsky, Thomas (1991), “Orientation of signed graphs”, European Journal of Combinatorics 12: 361–375, doi:10.1016/s0195-6698(13)80118-7. 85.8. REFERENCES 315

• Zelinka, Bohdan (1974), “Polar graphs and railway traﬃc”, Aplikace Matematiky 19: 169–176.

• Zelinka, Bohdan (1976a), “Isomorphisms of polar and polarized graphs”, Czechoslovak Mathematical Journal 26: 339–351.

• Zelinka, Bohdan (1976b), “Analoga of Menger’s theorem for polar and polarized graphs”, Czechoslovak Math- ematical Journal 26: 352–360. Chapter 86

Small-world network

Small-world network example Hubs are bigger than other nodes Average degree= 3.833 Average shortest path length = 1.803. Clusterization coeﬃcient = 0.522

A small-world network is a type of mathematical graph in which most nodes are not neighbors of one another, but most nodes can be reached from every other node by a small number of hops or steps. Speciﬁcally, a small-world network is deﬁned to be a network where the typical distance L between two randomly chosen nodes (the number of steps required) grows proportionally to the logarithm of the number of nodes N in the network, that is:[1]

L ∝ log N

316 317

Random graph Average degree = 2.833 Average shortest path length = 2.109. Clusterization coeﬃcient = 0.167

In the context of a social network, this results in the small world phenomenon of strangers being linked by a short chain of acquaintances. Many empirical graphs show the small-world effect, e.g., social networks, the underlying architecture of the Internet, wikis such as Wikipedia, and gene networks. A certain category of small-world networks were identified as a class of random graphs by Duncan Watts and Steven Strogatz in 1998.[2] They noted that graphs could be classified according to two independent structural features, namely the clustering coefficient, and average node-to-node distance (also known as average shortest path length). Purely random graphs, built according to the Erdős–Rényi (ER) model, exhibit a small average shortest path length (varying typically as the logarithm of the number of nodes) along with a small clustering coefficient. Watts and Strogatz measured that in fact many real-world networks have a small average shortest path length, but also a clustering coefficient significantly higher than expected by random chance. Watts and Strogatz then proposed a novel graph model, currently named the Watts and Strogatz model, with (i) a small average shortest path length, and (ii) a large clustering coefficient. The crossover in the Watts–Strogatz model between a “large world” (such as a lattice) and a small world was first described by Barthelemy and Amaral in 1999.[3] This work was followed by a large number of studies, including exact results (Barrat and Weigt, 1999; Dorogovtsev and Mendes; Barmpoutis and Murray, 2010). 318 CHAPTER 86. SMALL-WORLD NETWORK

86.1 Properties of small-world networks

Small-world networks tend to contain cliques, and near-cliques, meaning sub-networks which have connections between almost any two nodes within them. This follows from the defining property of a high clustering coefficient. Secondly, most pairs of nodes will be connected by at least one short path. This follows from the defining property that the mean-shortest path length be small. Several other properties are often associated with small-world networks. Typically there is an over-abundance of hubs - nodes in the network with a high number of connections (known as high degree nodes). These hubs serve as the common connections mediating the short path lengths between other edges. By analogy, the small-world network of airline flights has a small mean-path length (i.e. between any two cities you are likely to have to take three or fewer flights) because many flights are routed through hub cities. This property is often analyzed by considering the fraction of nodes in the network that have a particular number of connections going into them (the degree distribution of the network). Networks with a greater than expected number of hubs will have a greater fraction of nodes with high degree, and consequently the degree distribution will be enriched at high degree values. This is known colloquially as a fat-tailed distribution. Graphs of very different topology qualify as small-world networks as long as they satisfy the two definitional requirements above. Network small-worldness has been quantified by comparing clustering and path length of a given network to an equivalent random network with same degree on average.[4] Another method for quantifying network small-worldness utilizes the original definition of the small-world network comparing the clustering of a given network to an equivalent lattice network and its path length to an equivalent random network.[5] The small-world measure ( ω ) is defined as[6]

ω = Lr − C L Cl

R. Cohen and Havlin[7][8] showed analytically that scale-free networks are ultra-small worlds. In this case, due to hubs, the shortest paths become signiﬁcantly smaller and scale as

L ∝ log log N

86.2 Examples of small-world networks

Small-world properties are found in many real-world phenomena, including websites with navigation menus, food chains, electric power grids, metabolite processing networks, networks of brain neurons, voter networks, telephone call graphs, and social inﬂuence networks. Networks of connected proteins have small world properties such as power-law obeying degree distributions.[9] Sim- ilarly transcriptional networks, in which the nodes are genes, and they are linked if one gene has an up or down- regulatory genetic inﬂuence on the other, have small world network properties.[10]

86.3 Examples of non-small-world networks

Networks are less likely to have the small-world properties if links between nodes arise mainly from spatial or temporal proximity, because there may be no short path between two “distant” nodes. Being constrained to physical space or time, as in a subway system or road network, tends to impede the formation of particularly long links that are conducive to hub formation. In another example, the famous theory of "six degrees of separation" between people tacitly presumes that the domain of discourse is the set of people alive at any one time. The number of degrees of separation between Albert Einstein and Alexander the Great is almost certainly greater than 30 and this network does not have small-world properties. A similarly constrained network would be the “went to school with” network: if two people went to the same college ten years apart from one another, it is unlikely that they have acquaintances in common amongst the student body. Similarly, the number of relay stations through which a message must pass was not always small. In the days when the post was carried by hand or on horseback, the number of times a letter changed hands between its source and destination would have been much greater than it is today. The number of times a message changed hands in the 86.4. NETWORK ROBUSTNESS 319

days of the visual telegraph (circa 1800–1850) was determined by the requirement that two stations be connected by line-of-sight. Tacit assumptions, if not examined, can cause a bias in the literature on graphs in favor of finding small-world networks (an example of the file drawer effect resulting from the publication bias).

86.4 Network robustness

It is hypothesized by some researchers such as Barabási that the prevalence of small world networks in biological systems may reflect an evolutionary advantage of such an architecture. One possibility is that small-world networks are more robust to perturbations than other network architectures. If this were the case, it would provide an advantage to biological systems that are subject to damage by mutation or viral infection. In a small world network with a degree distribution following a power-law, deletion of a random node rarely causes a dramatic increase in mean-shortest path length (or a dramatic decrease in the clustering coefficient). This follows from the fact that most shortest paths between nodes flow through hubs, and if a peripheral node is deleted it is unlikely to interfere with passage between other peripheral nodes. As the fraction of peripheral nodes in a small world network is much higher than the fraction of hubs, the probability of deleting an important node is very low. For example, if the small airport in Sun Valley, Idaho was shut down, it would not increase the average number of flights that other passengers traveling in the United States would have to take to arrive at their respective destinations. However, if random deletion of a node hits a hub by chance, the average path length can increase dramatically. This can be observed annually when northern hub airports, such as Chicago’s O'Hare airport, are shut down because of snow; many people have to take additional flights. By contrast, in a random network, in which all nodes have roughly the same number of connections, deleting a random node is likely to increase the mean-shortest path length slightly but significantly for almost any node deleted. In this sense, random networks are vulnerable to random perturbations, whereas small-world networks are robust. However, small-world networks are vulnerable to targeted attack of hubs, whereas random networks cannot be targeted for catastrophic failure. Appropriately, viruses have evolved to interfere with the activity of hub proteins such as p53, thereby bringing about the massive changes in cellular behavior which are conducive to viral replication.

86.5 Construction of small-world networks

The main mechanism to construct small-world networks is the Watts–Strogatz mechanism. Small-world networks can also be introduced with time-delay,[11] which will not only produces fractals but also chaos[12] under the right conditions, or transition to chaos in dynamics networks.[13] Degree-Diameter graphs are constructed such that the number of neighbors each vertex in the network has is bounded, while the distance from any given vertex in the network to any other vertex (the diameter of the network) is minimized. Constructing such small-world networks is done as part of the effort to find graphs of order close to the Moore bound. Another way to construct a small world network from scratch is given in Barmpoutis et al.,[14] where a network with very small average distance and very large average clustering is constructed. A fast algorithm of constant complexity is given, along with measurements of the robustness of the resulting graphs. Depending on the application of each network, one can start with one such “ultra small-world” network, and then rewire some edges, or use several small such networks as subgraphs to a larger graph. Small-world properties can arise naturally in social networks and other real-world systems via the process of dual- phase evolution. This is particularly common where time or spatial constraints limit the addition of connections between vertices The mechanism generally involves periodic shifts between phases, with connections being added during a “global” phase and being reinforced or removed during a “local” phase. See also: Diffusion-limited aggregation, pattern formation

86.6 Applications 320 CHAPTER 86. SMALL-WORLD NETWORK

86.6.1 Applications to sociology

The advantages to small world networking for social movement groups are their resistance to change due to the filtering apparatus of using highly connected nodes, and its better effectiveness in relaying information while keeping the number of links required to connect a network to a minimum.[15] The small world network model is directly applicable to affinity group theory represented in sociological arguments by William Finnegan. Affinity groups are social movement groups that are small and semi-independent pledged to a larger goal or function. Though largely unaffiliated at the node level, a few members of high connectivity function as connectivity nodes, linking the different groups through networking. This small world model has proven an extremely effective protest organization tactic against police action.[16] Clay Shirky argues that the larger the social network created through small world networking, the more valuable the nodes of high connectivity within the network.[15] The same can be said for the affinity group model, where the few people within each group connected to outside groups allowed for a large amount of mobilization and adaptation. A practical example of this is small world networking through affinity groups that William Finnegan outlines in reference to the 1999 Seattle WTO protests.

86.6.2 Applications to earth sciences

Many networks studied in geology and geophysics have been shown to have characteristics of small-world networks. Networks deﬁned in fracture systems and porous substances have demonstrated these characteristics.[17] The seismic network in the Southern California region may be a small-world network.[18] The examples above occur on very diﬀerent spatial scales, demonstrating the scale invariance of the phenomenon in the earth sciences.

86.6.3 Applications to computing

Small-world networks have been used to estimate the usability of information stored in large databases. The measure is termed the Small World Data Transformation Measure.[19][20] The greater the database links align to a small-world network the more likely a user is going to be able to extract information in the future. This usability typically comes at the cost of the amount of information that can be stored in the same repository. The Freenet peer-to-peer network has been shown to form a small-world network in simulation,[21] allowing information to be stored and retrieved in a manner that scales eﬃciency as the network grows.

86.6.4 Small-world neural networks in the brain

Both anatomical connections in the brain[22] and the synchronization networks of cortical neurons[23] exhibit small- world topology. A small-world network of neurons can exhibit short-term memory. A computer model developed by Solla et al.[24][25] had two stable states, a property (called bistability) thought to be important in memory storage. An activating pulse generated self-sustaining loops of communication activity among the neurons. A second pulse ended this activity. The pulses switched the system between stable states: ﬂow (recording a “memory”), and stasis (holding it). On a more general level, many large-scale neural networks in the brain, such as the visual system and brain stem, exhibit small-world properties.[4]

86.7 Small world with a distribution of link length

The WS model includes a uniform distribution of long-range links. When the distribution of link lengths follows a power law distribution, the mean distance between two sites changes depending on the power of the distribution.[26]

86.8 See also

• Barabási–Albert model • Dual-phase evolution 86.9. REFERENCES 321

• Dunbar’s number

• Erdős number

• Erdős–Rényi (ER) model

• Scale-free network

• Six degrees of Kevin Bacon

• Small world experiment

• Social network

• Watts and Strogatz Model

86.9 References

[1] http://www.nature.com/nature/journal/v393/n6684/full/393440a0.html

[2] Watts, Duncan J.; Strogatz, Steven H. (June 1998). “Collective dynamics of 'small-world' networks”. Nature 393 (6684): 440–442. Bibcode:1998Natur.393..440W. doi:10.1038/30918. PMID 9623998. Papercore Summary http://www.papercore. org/Watts1998

[3] Barthelemy, M.; Amaral, LAN (1999). “Small-world networks: Evidence for a crossover picture”. Phys. Rev. Lett. 82 (15): 3180. arXiv:cond-mat/9903108. Bibcode:1999PhRvL..82.3180B. doi:10.1103/PhysRevLett.82.3180.

[4] The brainstem reticular formation is a small-world, not scale-free, network M. D. Humphries, K. Gurney and T. J. Prescott, Proc. Roy. Soc. B 2006 273, 503–511, doi:10.1098/rspb.2005.3354

[5] The ubiquity of small-world networks Q.K. Telesford, K.E. Joyce, S. Hayasaka, J.H. Burdette, P.J. Laurienti, Brain Con- nect. 2011;1(5):367–75, doi:10.1089/brain.2011.0038

[6] Telesford, Joyce, Hayasaka, Burdette, and Laurienti (2011). “The Ubiquity of Small-World Networks”. Brain Connectivity (0038).

[7] R. Cohen, S. Havlin, and D. ben-Avraham (2002). “Structural properties of scale free networks”. Handbook of graphs and networks (Wiley-VCH, 2002) (Chap. 4).

[8] R. Cohen, S. Havlin (2003). “Scale-free networks are ultrasmall”. Phys. Rev. Lett. 90 (5): 058701. arXiv:cond- mat/0205476. Bibcode:2003PhRvL..90e8701C. doi:10.1103/PhysRevLett.90.058701. PMID 12633404.

[9] Bork, P.; Jensen, LJ; von Mering, C.; Ramani, A.; Lee, I.; Marcotte, EM. (2004). “Protein interaction networks from yeast to human” (PDF). Current Opinion in Structural Biology 14 (3): 292–299. doi:10.1016/j.sbi.2004.05.003. PMID 15193308.

[10] Van Noort, V; Snel, B; Huynen, MA. (Mar 2004). “The yeast coexpression network has a small-world, scale-free architecture and can be explained by a simple model”. EMBO Rep. 5 (3): 280–4. doi:10.1038/sj.embor.7400090. PMC 1299002. PMID 14968131.

[11] X. S. Yang, Fractals in small-world networks with time-delay, Chaos, Solitons & Fractals, vol. 13, 215–219 (2002)

[12] X. S. Yang, Chaos in small-world networks, Phys. Rev. E 63, 046206 (2001)

[13] W. Yuan, X. S. Luo, P. Jiang, B. Wang, J. Fang, Transition to chaos in small-world dynamical network

[14] D.Barmpoutis and R.M. Murray (2010). “Networks with the Smallest Average Distance and the Largest Average Cluster- ing”. arXiv:1007.4031 [q-bio.MN].

[15] Shirky, Clay. 2008. Here Comes Everybody

[16] Finnegan, William “Aﬃnity Groups and the Movement Against Corporate Globalization”

[17] X. S. Yang, Small-world networks in geophysics, Geophys. Res. Lett., 28(13), 2549–2552 (2001)

[18] A. Jimenez, K. F. Tiampo, and A. M. Posadas, Small-world in a seismic network: the California case, Nonlin. Processes Geophys., 15, 389–395 (2008) 322 CHAPTER 86. SMALL-WORLD NETWORK

[19] http://mike2.openmethodology.org/wiki/Small_Worlds_Data_Transformation_Measure

[20] Hillard, Robert (2010). Information-Driven Business. Wiley. ISBN 978-0-470-62577-4.

[21] Sandberg, Oskar. “Searching in a Small World” (PDF).

[22] Sporns, Olaf; Chialvo DR; Kaiser M; Hilgetag CC (2004). “Organization, development and function of complex brain networks”. Trends Cogn Sci. 8 (9): 418–425. doi:10.1016/j.tics.2004.07.008. PMID 15350243.

[23] Yu, Shan; D. Huang; W. Singer; D. Nikolić (2008). “A Small World of Neuronal Synchrony”. Cerebral Cortex 18 (12): 2891–2901. doi:10.1093/cercor/bhn047. PMC 2583154. PMID 18400792.

[24] Cohen, Philip. Small world networks key to memory. New Scientist. 26 May 2004.

[25] Sara Solla’s Lecture & Slides: Self-Sustained Activity in a Small-World Network of Excitable Neurons

[26] D. Li, K. Kosmidis, A. Bunde, S. Havlin (2011). “Dimension of spatially embedded networks”. Nature Physics. Bibcode:2011NatPh...7..481D. doi:10.1038/nphys1932.

86.9.1 Books

• Buchanan, Mark (2003). Nexus: Small Worlds and the Groundbreaking Theory of Networks. Norton, W. W. & Company, Inc. ISBN 0-393-32442-7.

• Dorogovtsev, S.N. and Mendes, J.F.F. (2003). Evolution of Networks: from biological networks to the Internet and WWW. Oxford University Press. ISBN 0-19-851590-1.

• Watts, D. J. (1999). Small Worlds: The Dynamics of Networks Between Order and Randomness. Princeton University Press. ISBN 0-691-00541-9.

• Fowler, JH. (2005) “Turnout in a Small World,” in Alan Zuckerman, ed., Social Logic of Politics, Temple University Press, 269–287

• Reuven Cohen and Shlomo Havlin (2010). Complex Networks: Structure, Robustness and Function. Cambridge University Press.

86.9.2 Journal articles

• Albert, R.; Barabási A.L. (2002). “Statistical mechanics of complex networks”. Rev. Mod. Phys. 74: 47–97. arXiv:cond-mat/0106096. Bibcode:2002RvMP...74...47A. doi:10.1103/RevModPhys.74.47.

• Albert, R.; Barabási A.L. (1999). “Emergence of scaling in random networks”. Science 286 (5439): 509–12. arXiv:cond-mat/9910332. Bibcode:1999Sci...286..509B. doi:10.1126/science.286.5439.509. PMID 10521342.

• Barthelemy, M.; Amaral, LAN. (1999). “Small-world networks: Evidence for a crossover picture”. Phys. Rev. Lett. 82 (15): 3180. arXiv:cond-mat/9903108. Bibcode:1999PhRvL..82.3180B. doi:10.1103/PhysRevLett.82.3180.

• Dorogovtsev, S.N.; Mendes, J.F.F. (2000). “Exactly solvable analogy of small-world networks”. Europhys. Lett. 50: 1–7. arXiv:cond-mat/9907445. Bibcode:2000EL.....50....1D. doi:10.1209/epl/i2000-00227-1.

• Milgram, Stanley (1967). “The Small World Problem”. Psychology Today 1 (1): 60–67.

• Newman, Mark (2003). “The Structure and Function of Complex Networks”. SIAM Review 45 (2): 167–256. arXiv:cond-mat/0303516. Bibcode:2003SIAMR..45..167N. doi:10.1137/S003614450342480. pdf

• Ravid, D.; Rafaeli, S. (2004). “Asynchronous discussion groups as Small World and Scale Free Networks”. First Monday 9 (9).

• R. Parshani, S.V. Buldyrev, S. Havlin (2011). “Critical eﬀect of dependency groups on the function of networks”. PNAS 108: 1007. arXiv:1010.4498. Bibcode:2011PNAS..108.1007P. doi:10.1073/pnas.1008404108.

• S. V. Buldyrev, R. Parshani, G. Paul, H. E. Stanley, S. Havlin (2010). “Catastrophic cascade of failures in interdependent networks”. Nature 464 (7291): 1025–8. arXiv:0907.1182. Bibcode:2010Natur.464.1025B. doi:10.1038/nature08932. PMID 20393559. 86.10. EXTERNAL LINKS 323

86.10 External links

• Dynamic Proximity Networks by Seth J. Chandler, The Wolfram Demonstrations Project.

• Small-World Networks entry on Scholarpedia (by Mason A. Porter) Chapter 87

Snark (graph theory)

This article is about a term in graph theory. For other uses, see Snark (disambiguation). In the mathematical ﬁeld of graph theory, a snark is a connected, bridgeless cubic graph with chromatic index equal

The ﬂower snark J5 is one of six snarks on 20 vertices. to 4. In other words, it is a graph in which every vertex has three neighbors, and the edges cannot be colored by only

324 87.1. HISTORY 325 three colors without two edges of the same color meeting at a point. (By Vizing’s theorem, the chromatic index of a cubic graph is 3 or 4.) In order to avoid trivial cases, snarks are often restricted to have girth at least 5. Writing in The Electronic Journal of Combinatorics, Miroslav Chladný states that

87.1 History

P. G. Tait initiated the study of snarks in 1880, when he proved that the four color theorem is equivalent to the statement that no snark is planar.[2] The first known snark was the Petersen graph, discovered in 1898. In 1946, Croatian mathematician Danilo Blanuša discovered two more snarks, both on 18 vertices, now named the Blanuša snarks.[3] The fourth known snark was found two years later by Bill Tutte, under the pseudonym Blanche Descartes, and was a graph of order 210.[4][5] In 1973, George Szekeres found the fifth known snark — the Szekeres snark.[6] In 1975, Rufus Isaacs generalized Blanuša’s method to construct two infinite families of snarks: the flower snark and the BDS or Blanuša–Descartes–Szekeres snark, a family that includes the two Blanuša snarks, the Descartes snark and the Szekeres snark.[7] Isaacs also discovered a 30-vertices snark that does not belong to the BDS family and that is not a flower snark: the double-star snark. Snarks were so named by the American mathematician Martin Gardner in 1976, after the mysterious and elusive object of the poem The Hunting of the Snark by Lewis Carroll.[8]

87.2 Properties

All snarks are non-Hamiltonian, and many known snarks are hypohamiltonian: the removal of any single vertex leaves a Hamiltonian subgraph. A hypohamiltonian snark must be bicritical: the removal of any two vertices leaves a 3-edge-colorable subgraph.[9][10] It has been shown that the number of snarks for a given even number of vertices is bounded below by an exponential function.[11] (Being cubic graphs, all snarks must have an even number of vertices.) OEIS sequence A130315 contains the number of non-trivial snarks of 2n vertices for small values of n. The cycle double cover conjecture posits that in every bridgeless graph one can ﬁnd a collection of cycles covering each edge twice, or equivalently that the graph can be embedded onto a surface in such a way that all faces of the embedding are simple cycles. Snarks form the diﬃcult case for this conjecture: if it is true for snarks, it is true for all graphs.[12] In this connection, Branko Grünbaum conjectured that it was not possible to embed any snark onto a surface in such a way that all faces are simple cycles and such that every two faces either are disjoint or share only a single edge; however, a counterexample to Grünbaum’s conjecture was found by Martin Kochol.[13][14][15]

87.3 Snark theorem

W. T. Tutte conjectured that every snark has the Petersen graph as a minor. That is, he conjectured that the smallest snark, the Petersen graph, may be formed from any other snark by contracting some edges and deleting others. Equivalently (because the Petersen graph has maximum degree three) every snark has a subgraph that can be formed from the Petersen graph by subdividing some of its edges. This conjecture is a strengthened form of the four color theorem, because any graph containing the Petersen graph as a minor must be nonplanar. In 1999, Robertson, Sanders, Seymour, and Thomas announced a proof of this conjecture.[16] As of 2012, their proof remains largely unpublished.[17] See the Hadwiger conjecture for other problems and results relating graph coloring to graph minors. Tutte also conjectured a generalization of the snark theorem to arbitrary graphs: every bridgeless graph with no Petersen minor has a nowhere zero 4-ﬂow. That is, the edges of the graph may be assigned a direction, and a number from the set {1, 2, 3}, such that the sum of the incoming numbers minus the sum of the outgoing numbers at each vertex is divisible by four. As Tutte showed, for cubic graphs such an assignment exists if and only if the edges can be colored by three colors, so the conjecture follows from the snark theorem in this case. However, this conjecture remains open for graphs that need not be cubic.[18] 326 CHAPTER 87. SNARK (GRAPH THEORY)

87.4 List of snarks

• Petersen graph (10 vertices; discovered in 1898) • Tietze’s graph (12 vertices but with a girth of 3, generally not considered as a snark) • Blanuša snarks (two with 18 vertices; discovered in 1946) • Descartes snark (210 vertices; discovered by Bill Tutte in 1948) • Double-star snark (30 vertices) • Szekeres snark (50 vertices; discovered in 1973) • Watkins snark (50 vertices; discovered in 1989) • Flower snark (inﬁnite family on 20, 28, 36, 44... vertices; discovered in 1975)

A list of all of the snarks up to 36 vertices, except those with 36 vertices and girth 4, was generated by Gunnar Brinkmann, Jan Goedgebeur, Jonas Hägglund and Klas Markström in 2012.[19]

87.5 References

[1] Chladný, Miroslav; Škoviera, Martin (2010), “Factorisation of snarks”, The Electronic Journal of Combinatorics 17: R32. [2] Tait, P. G. (1880), “Remarks on the colourings of maps”, Proc. R. Soc. Edinburgh 10: 729 [3] Blanuša, Danilo (1946), “Problem četiriju boja”, Glasnik Mat. Fiz. Astr. Ser II 1: 31–42 [4] Blanche Descartes, Network-colourings, The Mathematical Gazette (London) 32, 67-69, 1948. [5] Martin Gardner, The Last Recreations: Hydras, Eggs, and Other Mathematical Mystifications, Springer, 2007, ISBN 0- 387-25827-2, ISBN 978-0-387-25827-0 [6] Szekeres, G. (1973), “Polyhedral decompositions of cubic graphs”, Bull. Austral. Math. Soc. 8 (3): 367–387, doi:10.1017/S0004972700042660. [7] Isaacs, R. (1975), “Infinite families of non-trivial trivalent graphs which are not Tait-colorable”, American Mathematical Monthly 82 (3): 221–239, doi:10.2307/2319844, JSTOR 2319844 [8] Gardner, Martin (1976), "Mathematical Games", Scientific American 4 (234): 126–130 [9] Steffen, E. (1998), “Classification and characterizations of snarks”, Discrete Mathematics 188 (1–3): 183–203, doi:10.1016/S0012- 365X(97)00255-0, MR 1630478 [10] Steffen, E. (2001), “On bicritical snarks”, Math. Slovaca 51 (2): 141–150, MR 1841443 [11] Skupień, Z. (2007), “6th Czech-Slovak International Symposium on Combinatorics, Graph Theory, Algorithms and Appli- cations”, Electronic Notes in Discrete Mathematics 28: 417–424, doi:10.1016/j.endm.2007.01.059 |contribution= ignored (help) [12] Jaeger, F. (1985), “A survey of the cycle double cover conjecture”, Annals of Discrete Mathematics 27 – Cycles in Graphs, North-Holland Mathematics Studies 27, pp. 1–12, doi:10.1016/S0304-0208(08)72993-1, ISBN 978-0-444-87803-8. [13] Kochol, Martin (1996), “Snarks without small cycles”, Journal of Combinatorial Theory, Series B 67, pp. 34–47. [14] Kochol, Martin (2009), “3-Regular non 3-edge-colorable graphs with polyhedral embeddings in orientable surfaces”, Graph Drawing 2008, Editors: I.G. Tollis, M. Patrignani, Lecture Notes in Computer Science 5417, pp. 319–323. [15] Kochol, Martin (2009), “Polyhedral embeddings of snarks in orientable surfaces”, Proceedings of the American Mathemat- ical Society 137, pp. 1613–1619. [16] Thomas, Robin (1999). “Recent Excluded Minor Theorems for Graphs”. Surveys in Combinatorics, 1999 (PDF). Cam- bridge University Press. pp. 201–222. [17] belcastro, sarah-marie (2012), “The continuing saga of snarks”, The College Mathematics Journal 43 (1): 82–87, doi:10.4169/college.math.j.43.1.082, MR 2875562. [18] 4-flow conjecture, Open Problem Garden. [19] Gunnar Brinkmann, Jan Goedgebeur, Jonas Hägglund and Klas Markström (2012), Generation and Properties of Snarks 87.6. EXTERNAL LINKS 327

87.6 External links

• Weisstein, Eric W., “Snark”, MathWorld.

• Alen Orbanić, Tomaž Pisanski, Milan Randić, and Brigite Servatius, "Blanuša Double", Mathematical Com- munications 9(2004),91-103. Chapter 88

Split graph

This article is about graphs that can be partitioned into a clique and an independent set. For cuts that form complete bipartite graphs, see split (graph theory). In graph theory, a branch of mathematics, a split graph is a graph in which the vertices can be partitioned into a clique

A split graph, partitioned into a clique and an independent set. and an independent set. Split graphs were ﬁrst studied by Földes and Hammer (1977a, 1977b), and independently introduced by Tyshkevich and Chernyak (1979).[1] A split graph may have more than one partition into a clique and an independent set; for instance, the path a–b–c is a split graph, the vertices of which can be partitioned in three diﬀerent ways:

1. the clique {a,b} and the independent set {c}

2. the clique {b,c} and the independent set {a}

3. the clique {b} and the independent set {a,c}

Split graphs can be characterized in terms of their forbidden induced subgraphs: a graph is split if and only if no induced subgraph is a cycle on four or ﬁve vertices, or a pair of disjoint edges (the complement of a 4-cycle).[2]

328 88.1. RELATION TO OTHER GRAPH FAMILIES 329

88.1 Relation to other graph families

From the definition, split graphs are clearly closed under complementation.[3] Another characterization of split graphs involves complementation: they are chordal graphs the complements of which are also chordal.[4] Just as chordal graphs are the intersection graphs of subtrees of trees, split graphs are the intersection graphs of distinct substars of star graphs.[5] Almost all chordal graphs are split graphs; that is, in the limit as n goes to infinity, the fraction of n-vertex chordal graphs that are split approaches one.[6] Because chordal graphs are perfect, so are the split graphs. The double split graphs, a family of graphs derived from split graphs by doubling every vertex (so the clique comes to induce an antimatching and the independent set comes to induce a matching), figure prominently as one of five basic classes of perfect graphs from which all others can be formed in the proof by Chudnovsky et al. (2006) of the Strong Perfect Graph Theorem. If a graph is both a split graph and an interval graph, then its complement is both a split graph and a comparability graph, and vice versa. The split comparability graphs, and therefore also the split interval graphs, can be characterized in terms of a set of three forbidden induced subgraphs.[7] The split cographs are exactly the threshold graphs, and the split permutation graphs are exactly the interval graphs that have interval graph complements.[8] Split graphs have cochromatic number 2.

88.2 Algorithmic problems

Let G be a split graph, partitioned into a clique C and an independent set I. Then every maximal clique in a split graph is either C itself, or the neighborhood of a vertex in I. Thus, it is easy to identify the maximum clique, and comple- mentarily the maximum independent set in a split graph. In any split graph, one of the following three possibilities must be true:[9]

1. There exists a vertex x in I such that C ∪ {x} is complete. In this case, C ∪ {x} is a maximum clique and I is a maximum independent set.

2. There exists a vertex x in C such that I ∪ {x} is independent. In this case, I ∪ {x} is a maximum independent set and C is a maximum clique.

3. C is a maximal clique and I is a maximal independent set. In this case, G has a unique partition (C,I) into a clique and an independent set, C is the maximum clique, and I is the maximum independent set.

Some other optimization problems that are NP-complete on more general graph families, including graph coloring, are similarly straightforward on split graphs. Hamiltonian Circuit remains NP-complete even for split graphs which are strongly chordal graphs.[10] It is also well known that the Minimum Dominating Set problem remains NP-complete for split graphs.[11]

88.3 Degree sequences

One remarkable property of split graphs is that they can be recognized solely from their degree sequences. Let the degree sequence of a graph G be d1 ≥ d2 ≥ ... ≥ dn, and let m be the largest value of i such that di ≥ i - 1. Then G is a split graph if and only if

∑m ∑n di = m(m − 1) + di. i=1 i=m+1

If this is the case, then the m vertices with the largest degrees form a maximum clique in G, and the remaining vertices constitute an independent set.[12] 330 CHAPTER 88. SPLIT GRAPH

88.4 Counting split graphs

Royle (2000) showed that n-vertex split graphs with n are in one-to-one correspondence with certain Sperner families. Using this fact, he determined a formula for the number of (nonisomorphic) split graphs on n vertices. For small values of n, starting from n = 1, these numbers are

1, 2, 4, 9, 21, 56, 164, 557, 2223, 10766, 64956, 501696, ... (sequence A048194 in OEIS).

This graph enumeration result was also proved earlier by Tyshkevich & Chernyak (1990).

88.5 Notes

[1] Földes & Hammer (1977a) had a more general definition, in which the graphs they called split graphs also included bipartite graphs (that is, graphs that be partitioned into two independent sets) and the complements of bipartite graphs (that is, graphs that can be partitioned into two cliques). Földes & Hammer (1977b) use the definition given here, which has been followed in the subsequent literature; for instance, it is Brandstädt, Le & Spinrad (1999), Definition 3.2.3, p.41.

[2] Földes & Hammer (1977a); Golumbic (1980), Theorem 6.3, p. 151.

[3] Golumbic (1980), Theorem 6.1, p. 150.

[4] Földes & Hammer (1977a); Golumbic (1980), Theorem 6.3, p. 151; Brandstädt, Le & Spinrad (1999), Theorem 3.2.3, p. 41.

[5] McMorris & Shier (1983); Voss (1985); Brandstädt, Le & Spinrad (1999), Theorem 4.4.2, p. 52.

[6] Bender, Richmond & Wormald (1985).

[7] Földes & Hammer (1977b); Golumbic (1980), Theorem 9.7, page 212.

[8] Brandstädt, Le & Spinrad (1999), Corollary 7.1.1, p. 106, and Theorem 7.1.2, p. 107.

[9] Hammer & Simeone (1981); Golumbic (1980), Theorem 6.2, p. 150.

[10] Müller (1996)

[11] Bertossi (1984)

[12] Hammer & Simeone (1981); Tyshkevich (1980); Tyshkevich, Melnikow & Kotov (1981); Golumbic (1980), Theorem 6.7 and Corollary 6.8, p. 154; Brandstädt, Le & Spinrad (1999), Theorem 13.3.2, p. 203. Merris (2003) further investigates the degree sequences of split graphs.

88.6 References

• Bender, E. A.; Richmond, L. B.; Wormald, N. C. (1985), “Almost all chordal graphs split”, J. Austral. Math. Soc.,A 38 (2): 214–221, doi:10.1017/S1446788700023077, MR 0770128. • Bertossi, Alan A. (1984), “Dominating sets for split and bipartite graphs”, Information Processing Letters 19: 37–40. • Brandstädt, Andreas; Le, Van Bang; Spinrad, Jeremy (1999), Graph Classes: A Survey, SIAM Monographs on Discrete Mathematics and Applications, ISBN 0-89871-432-X. • Chudnovsky, Maria; Robertson, Neil; Seymour, Paul; Thomas, Robin (2006), “The strong perfect graph theorem”, Annals of Mathematics 164 (1): 51–229, doi:10.4007/annals.2006.164.51, MR 2233847. • Földes, Stéphane; Hammer, Peter L. (1977a), “Split graphs”, Proceedings of the Eighth Southeastern Conference on Combinatorics, Graph Theory and Computing (Louisiana State Univ., Baton Rouge, La., 1977), Congressus Numerantium XIX, Winnipeg: Utilitas Math., pp. 311–315, MR 0505860. • Földes, Stéphane; Hammer, Peter L. (1977b), “Split graphs having Dilworth number two”, Canadian Journal of Mathematics 29 (3): 666–672, doi:10.4153/CJM-1977-069-1, MR 0463041. 88.7. FURTHER READING 331

• Golumbic, Martin Charles (1980), Algorithmic Graph Theory and Perfect Graphs, Academic Press, ISBN 0- 12-289260-7, MR 0562306. • Hammer, Peter L.; Simeone, Bruno (1981), “The splittance of a graph”, Combinatorica 1 (3): 275–284, doi:10.1007/BF02579333, MR 0637832. • McMorris, F. R.; Shier, D. R. (1983), “Representing chordal graphs on K₁,n", Commentationes Mathematicae Universitatis Carolinae 24: 489–494, MR 0730144. • Merris, Russell (2003), “Split graphs”, European Journal of Combinatorics 24 (4): 413–430, doi:10.1016/S0195- 6698(03)00030-1, MR 1975945. • Müller, Haiko (1996), “Hamiltonian Circuits in Chordal Bipartite Graphs”, Discrete Mathematics 156: 291– 298. • Royle, Gordon F. (2000), “Counting set covers and split graphs” (PDF), Journal of Integer Sequences 3 (2): 00.2.6, MR 1778996.

• Tyshkevich, Regina I. (1980), "[The canonical decomposition of a graph]", Doklady Akademii Nauk SSSR (in Russian) 24: 677–679, MR 0587712

• Tyshkevich, Regina I.; Chernyak, A. A. (1979), “Canonical partition of a graph deﬁned by the degrees of its vertices”, Isv. Akad. Nauk BSSR, Ser. Fiz.-Mat. Nauk (in Russian) 5: 14–26, MR 0554162.

• Tyshkevich, Regina I.; Chernyak, A. A. (1990), Еще один метод перечисления непомеченных комбина- торных объектов, Matem. Zametki (in Russian) 48 (6): 98–105, MR 1102626. Translated as “Yet another method of enumerating unmarked combinatorial objects” (1990), Mathematical notes of the Academy of Sci- ences of the USSR 48 (6): 1239–1245, doi:10.1007/BF01240267.

• Tyshkevich, Regina I.; Melnikow, O. I.; Kotov, V. M. (1981), “On graphs and degree sequences: the canonical decomposition”, Kibernetica (in Russian) 6: 5–8, MR 0689420.

• Voss, H.-J. (1985), “Note on a paper of McMorris and Shier”, Commentationes Mathematicae Universitatis Carolinae 26: 319–322, MR 0803929.

88.7 Further reading

• A chapter on split graphs appears in the book by Martin Charles Golumbic, “Algorithmic Graph Theory and Perfect Graphs”. Chapter 89

Squaregraph

A squaregraph.

In graph theory, a branch of mathematics, a squaregraph is a type of undirected graph that can be drawn in the plane in such a way that every bounded face is a quadrilateral and every vertex with three or fewer neighbors is incident to

332 89.1. RELATED GRAPH CLASSES 333 an unbounded face.

89.1 Related graph classes

The squaregraphs include as special cases trees, grid graphs, gear graphs, and the graphs of polyominos. As well as being planar graphs, squaregraphs are median graphs, meaning that for every three vertices u, v, and w there is a unique median vertex m(u,v,w) that lies on shortest paths between each pair of the three vertices.[1] As with median graphs more generally, squaregraphs are also partial cubes: their vertices can be labeled with binary strings such that the Hamming distance between strings is equal to the shortest path distance between vertices.

89.2 Characterization

Squaregraphs may be characterized in several ways other than via their planar embeddings:[2]

• They are the median graphs that do not contain as an induced subgraph any member of an inﬁnite family of forbidden graphs. These forbidden graphs are the cube (the simplex graph of K3), the Cartesian product of an edge and a claw K₁,₃ (the simplex graph of a claw), and the graphs formed from a gear graph by adding one more vertex connected to the hub of the wheel (the simplex graph of the disjoint union of a cycle with an isolated vertex).

• They are the graphs that are connected and bipartite, such that (if an arbitrary vertex r is picked as a root) every vertex has at most two neighbors closer to r, and such that at every vertex v, the link of v (a graph with a vertex for each edge incident to v and an edge for each 4-cycle containing v) is either a cycle of length greater than three or a disjoint union of paths.

• They are the dual graphs of arrangements of lines in the hyperbolic plane that do not have three mutually- crossing lines.

89.3 Algorithms

The characterization of squaregraphs in terms of distance from a root and links of vertices can be used together with breadth first search as part of a linear time algorithm for testing whether a given graph is a squaregraph, without any need to use the more complex linear-time algorithms for planarity testing of arbitrary graphs.[2] Several algorithmic problems on squaregraphs may be computed more efficiently than in more general planar or median graphs; for instance, Chepoi, Dragan & Vaxès (2002) and Chepoi, Fanciullini & Vaxès (2004) present linear time algorithms for computing the diameter of squaregraphs, and for finding a vertex minimizing the maximum distance to all other vertices.

89.4 Notes

[1] Soltan, Zambitskii & Prisǎcaru (1973). See Peterin (2006) for a discussion of planar median graphs more generally.

[2] Bandelt, Chepoi & Eppstein (2010).

89.5 References

• Bandelt, H.-J.; Chepoi, V.; Eppstein, D. (2010), “Combinatorics and geometry of ﬁnite and inﬁnite squaregraphs”, SIAM Journal on Discrete Mathematics 24 (4): 1399–1440, arXiv:0905.4537, doi:10.1137/090760301.

• Chepoi, V.; Dragan, F.; Vaxès, Y. (2002), “Center and diameter problem in planar quadrangulations and triangulations”, Proc. 13th Annu. ACM–SIAM Symp. on Discrete Algorithms (SODA 2002), pp. 346–355. 334 CHAPTER 89. SQUAREGRAPH

• Chepoi, V.; Fanciullini, C.; Vaxès, Y. (2004), “Median problem in some plane triangulations and quadrangulations”, Comput. Geom. 27 (3): 193–210, doi:10.1016/j.comgeo.2003.11.002. • Peterin, I. (2006), “A characterization of planar median graphs” (PDF), Discussiones Mathematicae Graph Theory 26: 41–48. • Soltan, P.; Zambitskii, D.; Prisǎcaru, C. (1973), Extremal Problems on Graphs and Algorithms of their Solution (In Russian), Chişinǎu, Moldova: Ştiinţa. Chapter 90

Strangulated graph

A strangulated graph, formed by using clique-sums to glue together a maximal planar graph (yellow) and two chordal graphs (red and blue). The red chordal graph can in turn be decomposed into clique-sums of four maximal planar graphs (two edges and two triangles).

In graph theoretic mathematics, a strangulated graph is a graph in which deleting the edges of any induced cycle of length greater than three would disconnect the remaining graph. That is, they are the graphs in which every peripheral cycle is a triangle.

90.1 Examples

In a maximal planar graph, or more generally in every polyhedral graph, the peripheral cycles are exactly the faces of a planar embedding of the graph, so a polyhedral graph is strangulated if and only if all the faces are triangles, or equivalently it is maximal planar. Every chordal graph is strangulated, because the only induced cycles in chordal graphs are triangles, so there are no longer cycles to delete.

335 336 CHAPTER 90. STRANGULATED GRAPH

90.2 Characterization

A clique-sum of two graphs is formed by identifying together two equal-sized cliques in each graph, and then possibly deleting some of the clique edges. For the version of clique-sums relevant to strangulated graphs, the edge deletion step is omitted. A clique-sum of this type between two strangulated graphs results in another strangulated graph, for every long induced cycle in the sum must be conﬁned to one side or the other (otherwise it would have a chord between the vertices at which it crossed from one side of the sum to the other), and the disconnected parts of that side formed by deleting the cycle must remain disconnected in the clique-sum. Every chordal graph can be decomposed in this way into a clique-sum of complete graphs, and every maximal planar graph can be decomposed into a clique-sum of 4-vertex-connected maximal planar graphs. As Seymour & Weaver (1984) show, these are the only possible building blocks of strangulated graphs: the strangulated graphs are exactly the graphs that can be formed as clique-sums of complete graphs and maximal planar graphs.

90.3 References

• Seymour, P. D.; Weaver, R. W. (1984), “A generalization of chordal graphs”, Journal of Graph Theory 8 (2): 241–251, doi:10.1002/jgt.3190080206, MR 742878. Chapter 91

Strongly chordal graph

In the mathematical area of graph theory, an undirected graph G is strongly chordal if it is a chordal graph and every cycle of even length (≥ 6) in G has an odd chord, i.e., an edge that connects two vertices that are an odd distance (>1) apart from each other in the cycle.[1]

91.1 Characterizations

Strongly chordal graphs have a forbidden subgraph characterization as the graphs that do not contain an induced cycle of length greater than three or an n-sun (n ≥ 3) as an induced subgraph.[2] An n-sun is a chordal graph with 2n vertices, partitioned into two subsets U = {u1, u2,...} and W = {w1, w2,...}, such that each vertex wi in W has exactly two neighbors, ui and u₍i ₊ ₁₎ ₒ n. An n-sun cannot be strongly chordal, because the cycle u1w1u2w2... has no odd chord. Strongly chordal graphs may also be characterized as the graphs having a strong perfect elimination ordering, an ordering of the vertices such that the neighbors of any vertex that come later in the ordering form a clique and such that, for each i < j < k < l, if the ith vertex in the ordering is adjacent to the kth and the lth vertices, and the jth and kth vertices are adjacent, then the jth and lth vertices must also be adjacent.[3] A graph is strongly chordal if and only if every one of its induced subgraphs has a simple vertex, a vertex whose neighbors have neighborhoods that are linearly ordered by inclusion.[4] Also, a graph is strongly chordal if and only if it is chordal and every cycle of length ﬁve or more has a 2-chord triangle, a triangle formed by two chords and an edge of the cycle.[5] A graph is strongly chordal if and only if each of its induced subgraphs is a dually chordal graph.[6] Strongly chordal graphs may also be characterized in terms of the number of complete subgraphs each edge partici- pates in.[7] Yet another characterization is given in.[8]

91.2 Recognition

It is possible to determine whether a graph is strongly chordal in polynomial time, by repeatedly searching for and removing a simple vertex. If this process eliminates all vertices in the graph, the graph must be strongly chordal; otherwise, if this process ﬁnds a subgraph without any more simple vertices, the original graph cannot be strongly chordal. For a strongly chordal graph, the order in which the vertices are removed by this process is a strong perfect elimination ordering.[9] Alternative algorithms are now known that can determine whether a graph is strongly chordal and, if so, construct a strong perfect elimination ordering more eﬃciently, in time O(min(n2,(n + m) log n)) for a graph with n vertices and m edges.[10]

337 338 CHAPTER 91. STRONGLY CHORDAL GRAPH

91.3 Subclasses

An important subclass (based on phylogeny) is the class of leaf powers deﬁned as follows: A tree T is a k-leaf root of a graph G=(V,E) if V is contained in the set of leaves of T and for vertices x,y in V, xy is an edge in E if and only if their distance in T is at most k. Then G is called a k-leaf power. A graph is a leaf power if it is a k-leaf power for some k.[11] Since powers of strongly chordal graphs are strongly chordal and trees are strongly chordal, we have: Leaf powers form a (proper) subclass of strongly chordal graphs. It is easy to see that a graph is a 2-leaf power if and only if it is [12] the disjoint union of cliques (i.e., a P3-free graph). Characterizations of 3- and 4-leaf powers are given in, which enable linear time recognition. For k > 5 the recognition problem of k-leaf powers is open, and likewise it is an open problem how to recognize leaf powers in polynomial time. Another important subclass of strongly chordal graphs are interval graphs. In [13] it is shown that interval graphs and the larger class of rooted directed path graphs are leaf powers.

91.4 Algorithmic problems

Since strongly chordal graphs are both chordal graphs and dually chordal graphs, various NP-complete problems such as Independent Set, Clique, Coloring, Clique Cover, Dominating Set, and Steiner Tree can be solved eﬃciently for strongly chordal graphs. Graph isomorphism is isomorphism-complete for strongly chordal graphs.[14] Hamiltonian Circuit remains NP-complete for strongly chordal split graphs.[15]

91.5 Notes

[1] Brandstädt, Le & Spinrad (1999), Deﬁnition 3.4.1, p. 43.

[2] Chang (1982); Farber (1983); Brandstädt, Le & Spinrad (1999), Theorem 7.2.1, p. 112.

[3] Farber (1983); Brandstädt, Le & Spinrad (1999), Theorem 5.5.1, p. 77.

[4] Farber (1983); Brandstädt, Le & Spinrad (1999), Theorem 5.5.2, p. 78.

[5] Dahlhaus, Manuel & Miller (1998).

[6] Brandstädt et al. (1998), Corollary 3, p. 444

[7] McKee (1999)

[8] De Caria & McKee (2014)

[9] Farber (1983).

[10] Lubiw (1987); Paige & Tarjan (1987); Spinrad (1993).

[11] Nishimura, Ragde & Thilikos (2002)

[12] Rautenbach (2006),Brandstädt & Le (2006),Brandstädt, Le & Sritharan (2008)

[13] Brandstädt et al. (2010)

[14] Uehara, Toda & Nagoya (2005)

[15] Müller (1996)

91.6 References

• Brandstädt, Andreas; Dragan, Feodor; Chepoi, Victor; Voloshin, Vitaly (1998), “Dually Chordal Graphs”, SIAM Journal on Discrete Mathematics 11: 437–455, doi:10.1137/s0895480193253415. • Brandstädt, Andreas; Hundt, Christian; Mancini, Federico; Wagner, Peter (2010), “Rooted directed path graphs are leaf powers”, Discrete Mathematics 310: 897–910. 91.6. REFERENCES 339

• Brandstädt, Andreas; Le, Van Bang (2006), “Structure and linear time recognition of 3-leaf powers”, Information Processing Letters 98: 133–138. • Brandstädt, Andreas; Le, Van Bang; Sritharan, R. (2008), “Structure and linear time recognition of 4-leaf powers”, ACM Transactions on Algorithms 5: Article 11. • Brandstädt, Andreas; Le, Van Bang; Spinrad, Jeremy (1999), Graph Classes: A Survey, SIAM Monographs on Discrete Mathematics and Applications, ISBN 0-89871-432-X. • Chang, G. J. (1982), K-domination and Graph Covering Problems, Ph.D. thesis, Cornell University.

• Dahlhaus, E.; Manuel, P. D.; Miller, M. (1998), “A characterization of strongly chordal graphs”, Discrete Mathematics 187 (1–3): 269–271, doi:10.1016/S0012-365X(97)00268-9.

• De Caria, P.; McKee, T.A. (2014), “Maxclique and unit disk characterizations of strongly chordal graphs”, Discussiones Mathematicae Graph Theory 34: 593–602, doi:10.7151/dmgt.1757.

• Farber, M. (1983), “Characterizations of strongly chordal graphs”, Discrete Mathematics 43 (2–3): 173–189, doi:10.1016/0012-365X(83)90154-1. • Lubiw, A. (1987), “Doubly lexical orderings of matrices”, SIAM Journal on Computing 16 (5): 854–879, doi:10.1137/0216057. • McKee, T. A. (1999), “A new characterization of strongly chordal graphs”, Discrete Mathematics 205 (1–3): 245–247, doi:10.1016/S0012-365X(99)00107-7. • Müller, H. (1996), “Hamiltonian Circuits in Chordal Bipartite Graphs”, Discrete Mathematics 156: 291–298.

• Nishimura, N.; Ragde, P.; Thilikos, D.M. (2002), “On graph powers for leaf-labeled trees”, Journal of Algo- rithms 42: 69–108.

• Paige, R.; Tarjan, R. E. (1987), “Three partition reﬁnement algorithms”, SIAM Journal on Computing 16 (6): 973–989, doi:10.1137/0216062.

• Rautenbach, D. (2006), “Some remarks about leaf roots”, Discrete Mathematics 306: 1456–1461. • Spinrad, J. (1993), “Doubly lexical ordering of dense 0–1 matrices”, Information Processing Letters 45 (2): 229–235, doi:10.1016/0020-0190(93)90209-R. • Uehara, R.; Toda, S.; Nagoya, T. (2005), “Graph isomorphism completeness for chordal bipartite and strongly chordal graphs”, Discrete Applied Mathematics 145 (3): 479–482, doi:10.1016/j.dam.2004.06.008. Chapter 92

Strongly regular graph

12 0 1 11 2 10 3

9 4 8 5 7 6

The Paley graph of order 13, a strongly regular graph with parameters srg(13,6,2,3).

In graph theory, a strongly regular graph is deﬁned as follows. Let G = (V,E) be a regular graph with v vertices and degree k. G is said to be strongly regular if there are also integers λ and μ such that:

340 92.1. PROPERTIES 341

• Every two adjacent vertices have λ common neighbours. • Every two non-adjacent vertices have μ common neighbours.

A graph of this kind is sometimes said to be an srg(v, k, λ, μ). Strongly regular graphs were introduced by Raj Chandra Bose in 1963.[1] Some authors exclude graphs which satisfy the deﬁnition trivially, namely those graphs which are the disjoint union of one or more equal-sized complete graphs,[2][3] and their complements, the Turán graphs. The complement of an srg(v, k, λ, μ) is also strongly regular. It is an srg(v, v−k−1, v−2−2k+μ, v−2k+λ). A strongly regular graph is a distance-regular graph with diameter 2, but only if μ is non-zero.

92.1 Properties

92.1.1 Relationship between Parameters

The four parameters in an srg(v, k, λ, μ) are not independent and must obey the following relation:

(v − k − 1)µ = k(k − λ − 1) The above relation can be derived very easily through a counting argument as follows:

• Imagine the nodes of the graph to lie in three levels. Pick any node as the root node, in Level 0. Then its k neighbor nodes lie in Level 1, and all other nodes lie in Level 2. • Nodes in Level 1 are directly connected to the root, hence they must have λ other neighbors in common with the root, and these common neighbors must also be in Level 1. Since each node has degree k, there are k − λ − 1 edges remaining for each Level 1 node to connect to nodes in Level 2. Therefore, there are k × (k − λ − 1) edges between Level 1 and Level 2. • Nodes in Level 2 are not directly connected to the root, hence they must have μ common neighbors with the root, and these common neighbors must all be in Level 1. There are (v − k − 1) nodes in Level 2, and each is connected to μ nodes in Level 1. Therefore the number of edges between Level 1 and Level 2 is (v −k−1)×µ . • Equating the two expressions for the edges between Level 1 and Level 2, the relation follows.

92.1.2 Adjacency Matrix

Let I denote the identity matrix (of order v) and let J denote the matrix whose entries all equal 1. The adjacency matrix A of a strongly regular graph satisﬁes two equations. First,

AJ = JA = kJ, which is a trivial restatement of the vertex degree requirement; incidentally, this shows that k is an eigenvalue of the adjacency matrix with an all-ones eigenvector. Second,

A2 + (µ − λ)A + (µ − k)I = µJ, which expresses the strong regularity condition. The ﬁrst term gives the number of 2-step paths from each vertex to all vertices, the second term the 1-step paths, and the third term the 0-step paths (so to speak). For the vertex pairs directly connected by an edge, the equation reduces to the number of such 2-step paths being equal to λ. For the vertex pairs not directly connected by an edge, the equation reduces to the number of such 2-step paths being equal to μ. For the trivial self-pairs, the equation reduces to the degree being equal to k. Conversely, a graph which is not a complete or null graph whose adjacency matrix satisﬁes both of the above conditions is a strongly regular graph.[4] 342 CHAPTER 92. STRONGLY REGULAR GRAPH

92.1.3 Eigenvalues

• The adjacency matrix of the graph has exactly three eigenvalues:

• k whose multiplicity is 1 (as seen above) [ ] [ √ ] − − • 1 (λ − µ) + (λ − µ)2 + 4(k − µ) whose multiplicity is 1 (v − 1) − √2k+(v 1)(λ µ) 2 2 (λ−µ)2+4(k−µ) [ ] [ √ ] − − • 1 (λ − µ) − (λ − µ)2 + 4(k − µ) whose multiplicity is 1 (v − 1) + √2k+(v 1)(λ µ) 2 2 (λ−µ)2+4(k−µ)

• As the multiplicities must be integers, their expressions provide further constraints on the values of v, k, μ, and λ, related to the so-called Krein conditions.

• Strongly regular graphs for which 2k + (v − 1)(λ − µ) = 0 are called conference graphs because of their connection with symmetric conference matrices. Their parameters reduce to

( ) 1 − 1 − 1 − srg v, 2 (v 1), 4 (v 5), 4 (v 1) .

• Strongly regular graphs for which 2k+(v−1)(λ−µ) ≠ 0 have integer eigenvalues with unequal multiplicities.

92.2 Examples

• The cycle of length 5 is an srg(5, 2, 0, 1).

• The Petersen graph is an srg(10, 3, 0, 1).

• The Clebsch graph is an srg(16, 5, 0, 2).

• The Shrikhande graph is an srg(16, 6, 2, 2) which is not a distance-transitive graph.

• The Line graph of generalized quadrangle GQ(2, 4) is an srg(27, 10, 1, 5). In fact every generalized quadrangle of order (s, t) gives a strongly regular graph in this way.

• The Schläﬂi graph is an srg(27, 16, 10, 8).[5]

• The Chang graphs are srg(28, 12, 6, 4).

• The Hoﬀman–Singleton graph is an srg(50, 7, 0, 1).

• The Sims-Gewirtz graph is an (56, 10, 0, 2).

• The M22 graph is an srg(77, 16, 0, 4).

• The Brouwer–Haemers graph is an srg(81, 20, 1, 6).

• The Higman–Sims graph is an srg(100, 22, 0, 6).

• The Local McLaughlin graph is an srg(162, 56, 10, 24).

• The Cameron graph is an srg(231, 30, 9, 3).

• The Paley graph of order q is an srg(q,(q − 1)/2, (q − 5)/4, (q − 1)/4).

• The n × n square rook’s graph is an srg(n2, 2n − 2, n − 2, 2).

A strongly regular graph is called primitive if both the graph and its complement are connected. All the above graphs are primitive, as otherwise μ=0 or μ=k. 92.3. SEE ALSO 343

92.2.1 Moore graphs

The strongly regular graphs with λ = 0 are triangle free. Apart from the complete graphs on less than 3 vertices and all complete bipartite graphs the seven listed above are the only known ones. Strongly regular graphs with λ = 0 and μ = 1 are Moore graphs with girth 5. Again the three graphs given above, with parameters (5, 2, 0, 1), (10, 3, 0, 1) and (50, 7, 0, 1), are the only known ones. The only other possible set of parameters yielding a Moore graph is (3250, 57, 0, 1); it is unknown if such a graph exists, and if so, whether or not it is unique.

92.3 See also

• Seidel adjacency matrix

• Partial geometry

92.4 Notes

[1] https://projecteuclid.org/euclid.pjm/1103035734, R. C. Bose, Strongly regular graphs, partial geometries and partially balanced designs, Paciﬁc J. Math 13 (1963) 389–419. (p. 122)

[2] Brouwer, Andries E; Haemers, Willem H. Spectra of Graphs. p. 101

[3] Godsil, Chris; Royle, Gordon. Algebraic Graph Theory. Springer-Verlag New York, 2001, p. 218.

[4] Cameron, P.J.; van Lint, J.H. (1991), Designs, Graphs, Codes and their Links, London Mathematical Society Student Texts 22, Cambridge University Press, p. 37, ISBN 978-0-521-42385-4

[5] Weisstein, Eric W., “Schläﬂi graph”, MathWorld.

92.5 References

• A.E. Brouwer, A.M. Cohen, and A. Neumaier (1989), Distance Regular Graphs. Berlin, New York: Springer- Verlag. ISBN 3-540-50619-5, ISBN 0-387-50619-5

• Chris Godsil and Gordon Royle (2004), Algebraic Graph Theory. New York: Springer-Verlag. ISBN 0-387- 95241-1

92.6 External links

• Eric W. Weisstein, Mathworld article with numerous examples.

• Gordon Royle, List of larger graphs and families. • Andries E. Brouwer, Parameters of Strongly Regular Graphs.

• Brendan McKay, Some collections of graphs. • Ted Spence, Strongly regular graphs on at most 64 vertices. Chapter 93

Subhamiltonian graph

In graph theory and graph drawing, a subhamiltonian graph is a subgraph of a planar Hamiltonian graph.[1][2]

93.1 Deﬁnition

A graph G is subhamiltonian if G is a subgraph of another graph aug(G) on the same vertex set, such that aug(G) is planar and contains a Hamiltonian cycle. For this to be true, G itself must be planar, and additionally it must be possible to add edges to G, preserving planarity, in order to create a cycle in the augmented graph that passes through each vertex exactly once. The graph aug(G) is called a Hamiltonian augmentation of G.[2] It would be equivalent to define G to be subhamiltonian if G is a subgraph of a Hamiltonian planar graph, without requiring this larger graph to have the same vertex set. That is, for this alternative definition, it should be possible to add both vertices and edges to G to create a Hamiltonian cycle. However, if a graph can be made Hamiltonian by the addition of vertices and edges it can also be made Hamiltonian by the addition of edges alone, so this extra freedom does not change the definition.[3] In a subhamiltonian graph, a subhamiltonian cycle is a cyclic sequence of vertices such that adding an edge between each consecutive pair of vertices in the sequence preserves the planarity of the graph. A graph is subhamiltonian if and only if it has a subhamiltonian cycle.[4]

93.2 History and applications

The class of subhamiltonian graphs (but not this name for them) was introduced by Bernhart & Kainen (1979), who proved that these are exactly the graphs with two-page book embeddings.[5] Subhamiltonian graphs and Hamiltonian augmentations have also been applied in graph drawing to problems involving embedding graphs onto universal point sets, simultaneous embedding of multiple graphs, and layered graph drawing.[2]

93.3 Related graph classes

Some classes of planar graphs are necessarily Hamiltonian, and therefore also subhamiltonian; these include the 4-connected planar graphs[6] and the Halin graphs.[7] Every planar graph with maximum degree at most four is subhamiltonian,[4] as is every planar graph with no separating triangles.[8] If the edges of an arbitrary planar graph are subdivided into paths of length two, the resulting subdivided graph is always subhamiltonian.[2]

344 93.4. REFERENCES 345

93.4 References

[1] Heath, Lenwood S. (1987), “Embedding outerplanar graphs in small books”, SIAM Journal on Algebraic and Discrete Methods 8 (2): 198–218, doi:10.1137/0608018, MR 881181.

[2] Di Giacomo, Emilio; Liotta, Giuseppe (2010), “The Hamiltonian augmentation problem and its applications to graph drawing”, WALCOM: Algorithms and Computation, 4th International Workshop, WALCOM 2010, Dhaka, Bangladesh, February 10-12, 2010, Proceedings, Lecture Notes in Computer Science 5942, Berlin: Springer, pp. 35–46, doi:10.1007/978-3-642- 11440-3_4, MR 2749626.

[3] For instance in a 2003 technical report "Book embeddings of graphs and a theorem of Whitney", Paul Kainen deﬁnes subhamiltonian graphs to be subgraphs of planar Hamiltonian graphs, without restriction on the vertex set of the augmentation, but writes that “in the deﬁnition of subhamiltonian graph, one can require that the extension only involve the inclusion of new edges.”

[4] Bekos, Michael A.; Gronemann, Martin; Raftopoulou, Chrysanthi N. (2014), “Two-page book embeddings of 4-planar graphs”, STACS, arXiv:1401.0684.

[5] Bernhart, Frank R.; Kainen, Paul C. (1979), “The book thickness of a graph”, Journal of Combinatorial Theory, Series B 27 (3): 320–331, doi:10.1016/0095-8956(79)90021-2.

[6] Nishizeki, Takao; Chiba, Norishige (2008), “Chapter 10. Hamiltonian Cycles”, Planar Graphs: Theory and Algorithms, Dover Books on Mathematics, Courier Dover Publications, pp. 171–184, ISBN 9780486466712.

[7] Cornuéjols, G.; Naddef, D.; Pulleyblank, W. R. (1983), “Halin graphs and the travelling salesman problem”, Mathematical Programming 26 (3): 287–294, doi:10.1007/BF02591867.

[8] Kainen, Paul C.; Overbay, Shannon (2007), “Extension of a theorem of Whitney”, Applied Mathematics Letters 20 (7): 835–837, doi:10.1016/j.aml.2006.08.019, MR 2314718. Chapter 94

Symmetric graph

The Petersen graph is a (cubic) symmetric graph. Any pair of adjacent vertices can be mapped to another by an automorphism, since any ﬁve-vertex ring can be mapped to any other.

In the mathematical ﬁeld of graph theory, a graph G is symmetric (or arc-transitive) if, given any two pairs of adjacent vertices u1—v1 and u2—v2 of G, there is an automorphism

346 94.1. EXAMPLES 347

f : V(G) → V(G)

such that

[1] f(u1) = u2 and f(v1) = v2.

In other words, a graph is symmetric if its automorphism group acts transitively upon ordered pairs of adjacent vertices (that is, upon edges considered as having a direction).[2] Such a graph is sometimes also called 1-arc-transitive[2] or flag-transitive.[3] [1] By definition (ignoring u1 and u2), a symmetric graph without isolated vertices must also be vertex transitive. Since the definition above maps one edge to another, a symmetric graph must also be edge transitive. However, an edge- transitive graph need not be symmetric, since a—b might map to c—d, but not to d—c. Semi-symmetric graphs, for example, are edge-transitive and regular, but not vertex-transitive. Every connected symmetric graph must thus be both vertex-transitive and edge-transitive, and the converse is true for graphs of odd degree.[3] However, for even degree, there exist connected graphs which are vertex-transitive and edge-transitive, but not symmetric.[4] Such graphs are called half-transitive.[5] The smallest connected half-transitive graph is Holt’s graph, with degree 4 and 27 vertices.[1][6] Confusingly, some authors use the term “symmetric graph” to mean a graph which is vertex-transitive and edge-transitive, rather than an arc-transitive graph. Such a definition would include half-transitive graphs, which are excluded under the definition above. A distance-transitive graph is one where instead of considering pairs of adjacent vertices (i.e. vertices a distance of 1 apart), the definition covers two pairs of vertices, each the same distance apart. Such graphs are automatically symmetric, by definition.[1] A t-arc is defined to be a sequence of t+1 vertices, such that any two consecutive vertices in the sequence are adjacent, and with any repeated vertices being more than 2 steps apart. A t-transitive graph is a graph such that the automorphism group acts transitively on t-arcs, but not on (t+1)-arcs. Since 1-arcs are simply edges, every symmetric graph of degree 3 or more must be t-transitive for some t, and the value of t can be used to further classify symmetric graphs. The cube is 2-transitive, for example.[1]

94.1 Examples

Combining the symmetry condition with the restriction that graphs be cubic (i.e. all vertices have degree 3) yields quite a strong condition, and such graphs are rare enough to be listed. The Foster census and its extensions provide such lists.[7] The Foster census was begun in the 1930s by Ronald M. Foster while he was employed by Bell Labs,[8] and in 1988 (when Foster was 92[1]) the then current Foster census (listing all cubic symmetric graphs up to 512 vertices) was published in book form.[9] The first thirteen items in the list are cubic symmetric graphs with up to 30 vertices[10][11] (ten of these are also distance transitive; the exceptions are as indicated): Other well known cubic symmetric graphs are the Dyck graph, the Foster graph and the Biggs–Smith graph. The ten distance-transitive graphs listed above, together with the Foster graph and the Biggs–Smith graph, are the only cubic distance-transitive graphs. Non-cubic symmetric graphs include cycle graphs (of degree 2), complete graphs (of degree 4 or more when there are 5 or more vertices), hypercube graphs (of degree 4 or more when there are 16 or more vertices), and the graphs formed by the vertices and edges of the octahedron, icosahedron, cuboctahedron, and icosidodecahedron. The Rado graph forms an example of a symmetric graph with infinitely many vertices and infinite degree.

94.2 Properties

The vertex-connectivity of a symmetric graph is always equal to the degree d.[3] In contrast, for vertex-transitive graphs in general, the vertex-connectivity is bounded below by 2(d+1)/3.[2] A t-transitive graph of degree 3 or more has girth at least 2(t–1). However, there are no ﬁnite t-transitive graphs of degree 3 or more for t ≥ 8. In the case of the degree being exactly 3 (cubic symmetric graphs), there are none for t ≥ 6. 348 CHAPTER 94. SYMMETRIC GRAPH

94.3 See also

• Algebraic graph theory

• Gallery of named graphs • Regular map

94.4 References

[1] Biggs, Norman (1993). Algebraic Graph Theory (2nd ed.). Cambridge: Cambridge University Press. pp. 118–140. ISBN 0-521-45897-8.

[2] Godsil, Chris; Royle, Gordon (2001). Algebraic Graph Theory. New York: Springer. p. 59. ISBN 0-387-95220-9.

[3] Babai, L (1996). “Automorphism groups, isomorphism, reconstruction”. In Graham, R; Grötschel, M; Lovász, L. Hand- book of Combinatorics. Elsevier.

[4] Bouwer, Z. “Vertex and Edge Transitive, But Not 1-Transitive Graphs.” Canad. Math. Bull. 13, 231–237, 1970.

[5] Gross, J.L. and Yellen, J. (2004). Handbook of Graph Theory. CRC Press. p. 491. ISBN 1-58488-090-2.

[6] Holt, Derek F. (1981). “A graph which is edge transitive but not arc transitive”. Journal of Graph Theory 5 (2): 201–204. doi:10.1002/jgt.3190050210..

[7] Marston Conder, Trivalent symmetric graphs on up to 768 vertices, J. Combin. Math. Combin. Comput, vol. 20, pp. 41–63

[8] Foster, R. M. “Geometrical Circuits of Electrical Networks.” Transactions of the American Institute of Electrical Engineers 51, 309–317, 1932.

[9] “The Foster Census: R.M. Foster’s Census of Connected Symmetric Trivalent Graphs”, by Ronald M. Foster, I.Z. Bouwer, W.W. Chernoﬀ, B. Monson and Z. Star (1988) ISBN 0-919611-19-2

[10] Biggs, p. 148

[11] Weisstein, Eric W., "Cubic Symmetric Graph", from Wolfram MathWorld.

94.5 External links

• Cubic symmetric graphs (The Foster Census). Data ﬁles for all cubic symmetric graphs up to 768 vertices, and some cubic graphs with up to 1000 vertices. Gordon Royle, updated February 2001, retrieved 2009-04-18.

• Trivalent (cubic) symmetric graphs on up to 2048 vertices. Marston Conder, August 2006, retrieved 2009-08- 20. Chapter 95

Table of simple cubic graphs

The connected 3-regular (cubic) simple graphs are listed for small vertex numbers.

95.1 Connectivity

The number of simple cubic graphs on 4, 6, 8, 10,... vertices is 1, 2, 5, 19,... (sequence A002851 in OEIS). A classification according to edge connectivity is made as follows: the 1-connected and 2-connected graphs are defined as usual. This leaves the other graphs in the 3-connected class because each 3-regular graph can be split by cutting all edges adjacent to any of the vertices. To refine this definition in the light of the algebra of coupling of angular momenta (see below), a subdivision of the 3-connected graphs is helpful. We shall call

• Non-trivially 3-connected those that can be split by 3 edge cuts into sub-graphs with at least two vertices remaining in each part

• Cyclically 4-connected—all those not 1-connected, not 2-connected, and not non-trivially 3-connected

This declares the numbers 3 and 4 in the fourth column of the tables below.

95.2 Pictures

Ball-and-stick models of the graphs in another column of the table show the vertices and edges in the style of images of molecular bonds. Comments on the individual pictures contain girth, diameter, Wiener index, Estrada index and Kirchhoff index. A Hamiltonian circuit (where present) is indicated by enumerating vertices along that path from 1 upwards. (The positions of the nodes have been defined by minimizing a pair potential defined by the squared difference of the Euclidean and graph theoretic distance, placed in a Molfile, then rendered by Jmol.)

95.3 LCF notation

The LCF notation is a notation by Joshua Lederberg, Coxeter and Frucht, for the representation of cubic graphs that are Hamiltonian. The two edges along the cycle adjacent to any of the vertices are not written down. Let v be the vertices of the graph and describe the Hamiltonian circle along the p vertices by the edge sequence v0v1,v1v2, ... ,v-₂v-₁,v-₁v0. Halting at a vertex vᵢ, there is one unique vertex v at a distance dᵢ joined by a chord with vᵢ,

j = i + di (mod p), 2 ≤ di ≤ p − 2.

349 350 CHAPTER 95. TABLE OF SIMPLE CUBIC GRAPHS

The vector [d0,d1, ..., d-₁] of the p integers is a suitable, although not unique, representation of the cubic Hamiltonian graph. This is augmented by two additional rules:

1. If a dᵢ >p/2, replace it by dᵢ-p;

2. avoid repetition of a sequence of dᵢ if these are periodic and replace them by an exponential notation.

Since the starting vertex of the path is of no importance, the numbers in the representation may be cyclically permuted. If a graph contains diﬀerent Hamiltonian circuits, one may select one of these to accommodate the notation. The same graph may have diﬀerent LCF notations, depending on precisely how the vertices are arranged. Often the anti-palindromic representations with

dp−1−i = −di (mod p), i = 0, 1, . . . p/2 − 1 are preferred (if they exist), and the redundant part is then replaced by ";-". The LCF notation [5,−9,7,−7,9,−5]4, for example, and would at that stage be condensed to [5,−9,7;-]4.

95.4 Table

95.4.1 4 nodes

95.4.2 6 nodes

95.4.3 8 nodes

95.4.4 10 nodes

95.4.5 12 nodes

The LCF entries are absent above if the graph has no Hamiltonian cycle, which is rare (see Tait’s conjecture). In this case a list of edges between pairs of vertices labeled 0 to n-1 in the third column serves as an identiﬁer.

95.5 Vector coupling coeﬃcients

Each 4-connected (in the above sense) simple cubic graph on 2n nodes defines a class of quantum mechanical 3n-j symbols. Roughly speaking, each vertex represents a 3-jm symbol, the graph is converted to a digraph by assigning signs to the angular momentum quantum numbers j, the vertices are labelled with a handedness representing the order of the three j (of the three edges) in the 3-jm symbol, and the graph represents a sum over the product of all these numbers assigned to the vertices. There are 1 (6j), 1 (9j), 2 (12j), 5 (15j), 18 (18j), 84 (21j), 607 (24j), 6100 (27j), 78824 (30j), 1195280 (33j), 20297600 (36j), 376940415 (39j) etc. of these (sequence A175847 in OEIS). If they are equivalent to certain vertex-induced binary trees (cutting one edge and finding a cut that splits the remaining graph into two trees), they are representations of recoupling coefficients, and are then also known as Yutsis graphs (sequence A111916 in OEIS).

95.6 See also

• 3-jm symbol

• 6-j symbol 95.7. REFERENCES 351

95.7 References

• Yutsis, A. P.; Levinson, I. B.; Vanagas, V. V.; Sen, A. (1962). Mathematical Apparatus of the theory of angular momentum. Israel program for scientiﬁc translations. Bibcode:1962mata.book.....Y. • Massot, J.-N.; El-Baz, E.; Lafoucriere, J. (1967). “A general graphical method for angular momentum”. Re- views of Modern Physics 39 (2): 288–305. Bibcode:1967RvMp...39..288M. doi:10.1103/RevModPhys.39.288.

• Bussemaker, F. C.; Cobeljic, S.; Cvetkovic, D. M. (1976). “Computer investigations of cubic graphs” (PDF). • Bussemaker, F. C.; Cobeljic, S.; Cvetkovic, D. M.; Seidel, J. J. (1977). “Cubic graphs on <=14 vertices”. J. Combin. Theory B 23: 234–235. doi:10.1016/0095-8956(77)90034-X. • Frucht, R. (1977). “A canonical representation of trivalent Hamiltonian graphs”. Journal of Graph Theory 1 (1): 45–60. doi:10.1002/jgt.3190010111. MR 0463029. • Clark, L.; Entringer, R. (1983). “Smallest maximally non-Hamiltonian graphs”. Per. Mathem. Hungar. 14 (1): 57–68. doi:10.1007/BF02023582. MR 0697357. • Wormald, N. C. (1985). “Enumeration of cyclically 4-connected cubic graphs”. Journal of Graph Theory 9: 563–573. doi:10.1002/jgt.3190090418. MR 0890248. • Bar-Shalom, A.; Klapisch, M. (1988). “NJGRAF - an eﬃcient program for calculation of general recoupling coeﬃcients by graphical analysis, compatible with NJSYM”. Comp. Phys. Comm. 50 (3): 375–393. Bibcode:1988CoPhC..50..375B. doi:10.1016/0010-4655(88)90192-0.

• Brinkmann, G. (1996). “Fast generation of cubic graphs”. Journal of Graph Theory 23 (2): 139–149. doi:10.1002/(SICI)1097-0118(199610)23:2<139::AID-JGT5>3.0.CO;2-U. MR 1408342.

• Fack, V.; Pitre, S. N.; Van der Jeugt, J. (1997). “Calculation of general recoupling coeﬃcients using graphical methods”. Comp. Phys. Comm. 101 (1–2): 155–170. Bibcode:1997CoPhC.101..155F. doi:10.1016/S0010- 4655(96)00170-1. • Danos, M.; Fano, U. (1998). “Graphical analysis of angular momentum for collision products”. Physics Reports 304 (4): 155–227. Bibcode:1998PhR...304..155D. doi:10.1016/S0370-1573(98)00020-9. • Meringer, M. (1999). “Fast generation of regular graphs and construction of cages”. Journal of Graph Theory 30 (2): 137–146. doi:10.1002/(SICI)1097-0118(199902)30:2<137::AID-JGT7>3.0.CO;2-G. MR 1665972. • Van Dyck, D.; Brinkmann, G.; Fack, V.; McKay, B. D. (2005). “To be or not to be Yutsis: Algorithms for the decision problem”. Comp. Phys. Comm. 173 (1–2): 61–70. Bibcode:2005CoPhC.173...61V. doi:10.1016/j.cpc.2005.07.008. MR 2179511. • Van Dyck, D.; Fack, V. (2007). “On the reduction of Yutsis graphs”. Disc. Math. 307 (11–12): 1506–1515. doi:10.1016/j.disc.2005.11.088. MR 2311125. • Aldred, R. E. L.; Van Dyck, D.; Brinkmann, G.; Fack, V.; McKay, B. D. (2009). “Graph structural properties of non-Yutsis graphs allowing fast recognition”. Disc. Math. 157 (2): 377–386. doi:10.1016/j.dam.2008.03.020. MR 2479811.

• Mathar, Richard J. (2011). “The Wigner graphs up to 12 Vertices”. arXiv:1109.2358.

` Chapter 96

Threshold graph

An example of a threshold graph.

In graph theory, a threshold graph is a graph that can be constructed from a one-vertex graph by repeated applications

352 96.1. ALTERNATIVE DEFINITIONS 353

of the following two operations:

1. Addition of a single isolated vertex to the graph.

2. Addition of a single dominating vertex to the graph, i.e. a single vertex that is connected to all other vertices.

For example, the graph of the ﬁgure is a threshold graph. It can be constructed by beginning with a single-vertex graph (vertex 1), and then adding black vertices as isolated vertices and red vertices as dominating vertices, in the order in which they are numbered. Threshold graphs were ﬁrst introduced by Chvátal & Hammer (1977). A chapter on threshold graphs appears in Golumbic (1980), and the book Mahadev & Peled (1995) is devoted to them.

96.1 Alternative deﬁnitions

An equivalent definition is the following: a graph is a threshold graph if there are a real number S and for each vertex v a real vertex weight w(v) such that for any two vertices v, u , uv is an edge if and only if w(u) + w(v) > S . Another equivalent definition is this: a graph is a threshold graph if there are a real number T ∑and for each vertex v ⊆ ≤ a real vertex weight a(v) such that for any vertex set X V , X is independent if and only if v∈X a(v) T. The name “threshold graph” comes from these definitions: S is the “threshold” for the property of being an edge, or equivalently T is the threshold for being independent.

96.2 Decomposition

From the definition which uses repeated addition of vertices, one can derive an alternative way of uniquely describing a threshold graph, by means of a string of symbols. ϵ is always the first character of the string, and represents the first vertex of the graph. Every subsequent character is either u , which denotes the addition of an isolated vertex (or union vertex), or j , which denotes the addition of a dominating vertex (or join vertex). For example, the string ϵuuj represents a star graph with three leaves, while ϵuj represents a path on three vertices. The graph of the figure can be represented as ϵuuujuuj

96.3 Related classes of graphs and recognition

Threshold graphs are a special case of cographs, split graphs, and trivially perfect graphs. Every graph that is both a cograph and a split graph is a threshold graph. Every graph that is both a trivially perfect graph and the complementary graph of a trivially perfect graph is a threshold graph. Threshold graphs are also a special case of interval graphs. All these relations can be explained in terms of their characterisation by forbidden induced subgraphs. A cograph is a graph with no induced path on four vertices, P4, and a threshold graph is a graph with no induced P4,C4 nor 2K2.C4 is a cycle of four vertices and 2K2 is its complement, that is, two disjoint edges. This also explains why threshold graphs are closed under taking complements; the P4 is self-complementary, hence if a graph is P4-, C4- and 2K2-free, its complement is as well. Heggernes et al. showed that threshold graphs can be recognized in linear time; if a graph is not threshold, an obstruction (one of P4,C4, or 2K2) will be output.

96.4 See also

• Indiﬀerence graph

• Series-parallel graph 354 CHAPTER 96. THRESHOLD GRAPH

96.5 References

• Chvátal, Václav; Hammer, Peter L. (1977), “Aggregation of inequalities in integer programming”, in Hammer, P. L.; Johnson, E. L.; Korte, B. H.; et al., Studies in Integer Programming (Proc. Worksh. Bonn 1975), Annals of Discrete Mathematics 1, Amsterdam: North-Holland, pp. 145–162.

• Golumbic, Martin Charles (1980), Algorithmic Graph Theory and Perfect Graphs, New York: Academic Press. 2nd edition, Annals of Discrete Mathematics, 57, Elsevier, 2004.

• Mahadev, N. V. R.; Peled, Uri N. (1995), Threshold Graphs and Related Topics, Elsevier.

96.6 External links

• Threshold graphs, Information System on Graph Classes and their Inclusions. Chapter 97

Toroidal graph

A cubic graph with 14 vertices embedded on a torus

In mathematics, a toroidal graph is a graph that can be embedded on a torus. In other words, the graph’s vertices can be placed on a torus such that no edges cross.

355 356 CHAPTER 97. TOROIDAL GRAPH

97.1 Examples

The Heawood graph, the complete graph K7 (and hence K5 and K6), the Petersen graph (and hence the complete bipartite graph K₃,₃, since the Petersen graph contains a subdivision of it), one of the Blanuša snarks,[1] and all Möbius ladders are toroidal. More generally, any graph with crossing number 1 is toroidal. Some graphs with greater crossing numbers are also toroidal: the Möbius–Kantor graph, for example, has crossing number 4 and is toroidal.[2]

97.2 Properties

[3] Any toroidal graph has chromatic number at most 7. The complete graph K7 provides an example of toroidal graph with chromatic number 7.[4] Any triangle-free toroidal graph has chromatic number at most 4.[5] By a result analogous to Fáry’s theorem, any toroidal graph may be drawn with straight edges in a rectangle with periodic boundary conditions.[6] Furthermore, the analogue of Tutte’s spring theorem applies in this case.[7] Toroidal graphs also have book embeddings with at most 7 pages.[8]

97.3 See also

• Planar graph • Topological graph theory • Császár polyhedron

97.4 Notes

[1] Orbanić et al. (2004).

[2] Marušič & Pisanski (2000).

[3] Heawood (1890).

[4] Chartrand & Zhang (2008).

[5] Kronk & White (1972).

[6] Kocay, Neilson & Szypowski (2001).

[7] Gortler, Gotsman & Thurston (2006).

[8] Endo (1997).

97.5 References

• Chartrand, Gary; Zhang, Ping (2008), Chromatic graph theory, CRC Press, ISBN 978-1-58488-800-0. • Endo, Toshiki (1997), “The pagenumber of toroidal graphs is at most seven”, Discrete Mathematics 175 (1–3): 87–96, doi:10.1016/S0012-365X(96)00144-6, MR 1475841. • Gortler, Steven J.; Gotsman, Craig; Thurston, Dylan (2006), “Discrete one-forms on meshes and applications to 3D mesh parameterization”, Computer Aided Geometric Design 23 (2): 83–112, doi:10.1016/j.cagd.2005.05.002, MR 2189438. • Heawood, P. J. (1890), “Map colouring theorems”, Quarterly J. Math. Oxford Ser. 24: 322–339. • Kocay, W.; Neilson, D.; Szypowski, R. (2001), “Drawing graphs on the torus” (PDF), Ars Combinatoria 59: 259–277, MR 1832459. 97.5. REFERENCES 357

• Kronk, Hudson V.; White, Arthur T. (1972), “A 4-color theorem for toroidal graphs”, Proceedings of the Amer- ican Mathematical Society (American Mathematical Society) 34 (1): 83–86, doi:10.2307/2037902, JSTOR 2037902, MR 0291019.

• Marušič, Dragan; Pisanski, Tomaž (2000), “The remarkable generalized Petersen graph G(8,3)", Math. Slo- vaca 50: 117–121.

• Neufeld, Eugene; Myrvold, Wendy (1997), “Practical toroidality testing”, Proceedings of the Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 574–580.

• Orbanić, Alen; Pisanski, Tomaž; Randić, Milan; Servatius, Brigitte (2004), “Blanuša double”, Math. Commun. 9 (1): 91–103. Chapter 98

Trellis (graph)

Convolutional code trellis diagram

A trellis is a graph whose nodes are ordered into vertical slices (time), and with each node at each time connected to at least one node at an earlier and at least one node at a later time. The earliest and latest times in the trellis have only one node. Trellises are used in encoders and decoders for communication theory and encryption. They are also the central datatype used in Baum–Welch algorithm for Hidden Markov Models.

98.1 See also

• Trellis modulation • Trellis quantization

358 Chapter 99

Triangle-free graph

In the mathematical area of graph theory, a triangle-free graph is an undirected graph in which no three vertices form a triangle of edges. Triangle-free graphs may be equivalently deﬁned as graphs with clique number ≤ 2, graphs with girth ≥ 4, graphs with no induced 3-cycle, or locally independent graphs.

The triangle-free graphs with the most edges for their vertices are balanced complete bipartite graphs

By Turán’s theorem, the n-vertex triangle-free graph with the maximum number of edges is a complete bipartite graph in which the numbers of vertices on each side of the bipartition are as equal as possible.

99.1 Triangle ﬁnding problem

The triangle finding problem is the problem of determining whether a graph is triangle-free or not. When the graph does contain a triangle, algorithms are often required to output three vertices which form a triangle in the graph. It is possible to test whether a graph with m edges is triangle-free in time O(m1.41).[1] Another approach is to find the trace of A3, where A is the adjacency matrix of the graph. The trace is zero if and only if the graph is triangle-free. For dense graphs, it is more efficient to use this simple algorithm which relies on matrix multiplication, since it gets the time complexity down to O(n2.373), where n is the number of vertices. As Imrich, Klavžar & Mulder (1999) show, triangle-free graph recognition is equivalent in complexity to median graph recognition; however, the current best algorithms for median graph recognition use triangle detection as a subroutine rather than vice versa. The decision tree complexity or query complexity of the problem, where the queries are to an oracle which stores the

359 360 CHAPTER 99. TRIANGLE-FREE GRAPH

adjacency matrix of a graph, is Θ(n2). However, for quantum algorithms, the best known lower bound is Ω(n), but the best known algorithm is O(n35/27).[2]

99.2 Independence number and Ramsey theory

An independent set of √n vertices in an n-vertex triangle-free graph is easy to ﬁnd: either there is a vertex with greater than √n neighbors (in which case those neighbors are an independent set) or all vertices have fewer than √n neighbors [3] (in which case any maximal independent set must have at least √√n vertices). This bound can be tightened slightly: in every triangle-free graph there√ exists an independent set of Ω( n log n) vertices, and in some triangle-free graphs every independent set has O( n log n) vertices.[4] One way to generate triangle-free graphs in which all independent sets are small is the triangle-free process[5] in which one generates a maximal triangle-free graph by repeatedly adding randomly chosen edges that√ do not complete a triangle. With high probability, this process produces a graph with independence number O( n log n) . It is also possible to ﬁnd regular graphs with the same properties.[6]

t2 These results may also be interpreted as giving asymptotic bounds on the Ramsey numbers R(3,t) of the form Θ( log t ) t2 : if the edges of a complete graph on Ω( log t ) vertices are colored red and blue, then either the red graph contains a triangle or, if it is triangle-free, then it must have an independent set of size t corresponding to a clique of the same size in the blue graph.

99.3 Coloring triangle-free graphs

Much research about triangle-free graphs has focused on graph coloring. Every bipartite graph (that is, every 2-colorable graph) is triangle-free, and Grötzsch’s theorem states that every triangle-free planar graph may be 3- colored.[7] However, nonplanar triangle-free graphs may require many more than three colors. Mycielski (1955) deﬁned a construction, now called the Mycielskian, for forming a new triangle-free graph from another triangle-free graph. If a graph has chromatic number k, its Mycielskian has chromatic number k + 1, so this construction may be used to show that arbitrarily large numbers of colors may be needed to color nonplanar triangle- free graphs. In particular the Grötzsch graph, an 11-vertex graph formed by repeated application of Mycielski’s construction, is a triangle-free graph that cannot be colored with fewer than four colors, and is the smallest graph with this property.[8] Gimbel & Thomassen (2000) and Nilli (2000) showed that the number of colors needed to color any m-edge triangle-free graph is

( ) m1/3 O (log m)2/3 and that there exist triangle-free graphs that have chromatic numbers proportional to this bound. There have also been several results relating coloring to minimum degree in triangle-free graphs. Andrásfai, Erdős & Sós (1974) proved that any n-vertex triangle-free graph in which each vertex has more than 2n/5 neighbors must be bipartite. This is the best possible result of this type, as the 5-cycle requires three colors but has exactly 2n/5 neighbors per vertex. Motivated by this result, Erdős & Simonovits (1973) conjectured that any n-vertex triangle- free graph in which each vertex has at least n/3 neighbors can be colored with only three colors; however, Häggkvist (1981) disproved this conjecture by ﬁnding a counterexample in which each vertex of the Grötzsch graph is replaced by an independent set of a carefully chosen size. Jin (1995) showed that any n-vertex triangle-free graph in which each vertex has more than 10n/29 neighbors must be 3-colorable; this is the best possible result of this type, because Häggkvist’s graph requires four colors and has exactly 10n/29 neighbors per vertex. Finally, Brandt & Thomassé (2006) proved that any n-vertex triangle-free graph in which each vertex has more than n/3 neighbors must be 4- colorable. Additional results of this type are not possible, as Hajnal[9] found examples of triangle-free graphs with arbitrarily large chromatic number and minimum degree (1/3 − ε)n for any ε > 0.

99.4 Sensitivity and Block Sensitivity

For a Boolean function, the sensitivity of f at x , denoted s(f, x) , is the number of single-bit changes in x that change the value of f(x) . The sensitivity is then deﬁned to be the maximum value of the sensitivity at x across all values of x . 99.5. SEE ALSO 361

The Grötzsch graph is a triangle-free graph that cannot be colored with fewer than four colors

The block sensitivity, bs(f) , is likewise defined by looking at flipping multiple bits simultaneously.[10] Although most commonly examined boolean functions satisfy bs(f) = O(s(f)) , the Sensitivity Conjecture that bs(f) = O(s(f)2) has proven to be difficult to prove, causing mathematicians to consider the question of constructing examples of functions that exhibit large gaps between the two quantities.[10] Biderman et al. (2015) found the largest known gap between the two quantities by considering a function that is the indicator function for the property of being an isolated triangle-free graph. A triangle in a graph G is said to be isolated if the vertices comprising the triangle are adjacent to no other vertices besides those that comprise the triangle. Biderman et al. (2015) showed further that this result holds when “triangle” is replaced with any cycle or any clique.

99.5 See also

• Henson graph, an inﬁnite triangle-free graph that contains all ﬁnite triangle-free graphs as induced subgraphs

• Monochromatic triangle problem, the problem of partitioning the edges of a given graph into two triangle-free graphs 362 CHAPTER 99. TRIANGLE-FREE GRAPH

99.6 References

Notes

[1] Alon, Yuster & Zwick (1994).

[2] Lee, Magniez & Santha (2013), improving a previous algorithm by Belovs (2012).

[3] Boppana & Halldórsson (1992) p. 184, based on an idea from an earlier coloring approximation algorithm of Avi Wigder- son.

[4] Kim (1995).

[5] Erdős, Suen & Winkler (1995); Bohman (2009).

[6] Alon, Ben-Shimon & Krivelevich (2010).

[7] Grötzsch (1959); Thomassen (1994)).

[8] Chvátal (1974).

[9] see Erdős & Simonovits (1973).

[10] Hatami, Kulkarni & Pankratov (2010).

Sources

• Alon, Noga; Ben-Shimon, Sonny; Krivelevich, Michael (2010), “A note on regular Ramsey graphs”, Journal of Graph Theory 64 (3): 244–249, arXiv:0812.2386, doi:10.1002/jgt.20453, MR 2674496.

• Alon, N.; Yuster, R.; Zwick, U. (1994), “Finding and counting given length cycles”, Proceedings of the 2nd European Symposium on Algorithms, Utrecht, The Netherlands, pp. 354–364.

• Andrásfai, B.; Erdős, P.; Sós, V. T. (1974), “On the connection between chromatic number, maximal clique and minimal degree of a graph”, Discrete Mathematics 8 (3): 205–218, doi:10.1016/0012-365X(74)90133-2.

• Belovs, Aleksandrs (2012), “Span programs for functions with constant-sized 1-certiﬁcates”, Proceedings of the Forty-fourth Annual ACM Symposium on Theory of Computing (STOC '12), New York, NY, USA: ACM, pp. 77–84, doi:10.1145/2213977.2213985, ISBN 978-1-4503-1245-5.

• Biderman, Stella; Cuddy, Kevin; Li, Ang; Song, Min Jae (2015), On the Sensitivity of k-Uniform Hypergraph Properties, arXiv:1510.00354.

• Bohman, Tom (2009), “The triangle-free process”, Advances in Mathematics 221 (5): 1653–1677, arXiv:0806.4375, doi:10.1016/j.aim.2009.02.018, MR 2522430.

• Boppana, Ravi; Halldórsson, Magnús M. (1992), “Approximating maximum independent sets by excluding subgraphs”, BIT 32 (2): 180–196, doi:10.1007/BF01994876, MR 1172185.

• Brandt, S.; Thomassé, S. (2006), Dense triangle-free graphs are four-colorable: a solution to the Erdős-Simonovits problem (PDF).

• Chiba, N.; Nishizeki, T. (1985), “Arboricity and subgraph listing algorithms”, SIAM Journal on Computing 14 (1): 210–223, doi:10.1137/0214017.

• Chvátal, Vašek (1974), “The minimality of the Mycielski graph”, Graphs and combinatorics (Proc. Capital Conf., George Washington Univ., Washington, D.C., 1973), Lecture Notes in Mathematics 406, Springer- Verlag, pp. 243–246.

• Erdős, P.; Simonovits, M. (1973), “On a valence problem in extremal graph theory”, Discrete Mathematics 5 (4): 323–334, doi:10.1016/0012-365X(73)90126-X.

• Erdős, P.; Suen, S.; Winkler, P. (1995), “On the size of a random maximal graph”, Random Structures and Algorithms 6 (2–3): 309–318, doi:10.1002/rsa.3240060217. 99.7. EXTERNAL LINKS 363

• Gimbel, John; Thomassen, Carsten (2000), “Coloring triangle-free graphs with ﬁxed size”, Discrete Mathemat- ics 219 (1-3): 275–277, doi:10.1016/S0012-365X(00)00087-X. • Grötzsch, H. (1959), “Zur Theorie der diskreten Gebilde, VII: Ein Dreifarbensatz für dreikreisfreie Netze auf der Kugel”, Wiss. Z. Martin-Luther-U., Halle-Wittenberg, Math.-Nat. Reihe 8: 109–120. • Häggkvist, R. (1981), “Odd cycles of speciﬁed length in nonbipartite graphs”, Graph Theory (Cambridge, 1981), pp. 89–99. • Hatami, Pooya; Kulkarni, Raghav; Pankratov, Denis (2010), Variations on the Sensitivity Conjecture, arXiv:1011.0354.

• Itai, A.; Rodeh, M. (1978), “Finding a minimum circuit in a graph”, SIAM Journal on Computing 7 (4): 413– 423, doi:10.1137/0207033.

• Jin, G. (1995), “Triangle-free four-chromatic graphs”, Discrete Mathematics 145 (1-3): 151–170, doi:10.1016/0012- 365X(94)00063-O. • t2 Kim, J. H. (1995), “The Ramsey number R(3, t) has order of magnitude log t ", Random Structures and Algo- rithms (3 ed.) 7: 173–207, doi:10.1002/rsa.3240070302 delete character in |title= at position 19 (help). • Lee, Troy; Magniez, Frédéric; Santha, Miklos (2013), “Improved quantum query algorithms for triangle ﬁnd- ing and associativity testing”, Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2013), New Orleans, Louisiana, pp. 1486–1502, ISBN 978-1-611972-51-1.

• Mycielski, J. (1955), “Sur le coloriage des graphes”, Colloq. Math. 3: 161–162. • Nilli, A. (2000), “Triangle-free graphs with large chromatic numbers”, Discrete Mathematics 211 (1–3): 261– 262, doi:10.1016/S0012-365X(99)00109-0. • Shearer, J. B. (1983), “Note on the independence number of triangle-free graphs”, Discrete Mathematics 46 (1): 83–87, doi:10.1016/0012-365X(83)90273-X. • Thomassen, C. (1994), “Grötzsch’s 3-color theorem”, Journal of Combinatorial Theory, Series B 62 (2): 268– 279, doi:10.1006/jctb.1994.1069.

99.7 External links

• “Graphclass: triangle-free”, Information System on Graph Classes and their Inclusions Chapter 100

Trivially perfect graph

e d g h b c f a

e e

d g h d g h

b c f b c f

a a

Construction of a trivially perfect graph from nested intervals and from the reachability relationship in a tree

In graph theory, a trivially perfect graph is a graph with the property that in each of its induced subgraphs the size of the maximum independent set equals the number of maximal cliques.[1] Trivially perfect graphs were ﬁrst studied by (Wolk 1962, 1965) but were named by Golumbic (1978); Golumbic writes that “the name was chosen since it is trivial to show that such a graph is perfect.” Trivially perfect graphs are also known as comparability graphs of trees,[2] arborescent comparability graphs,[3] and quasi-threshold graphs.[4]

100.1 Equivalent characterizations

Trivially perfect graphs have several other equivalent characterizations:

• They are the comparability graphs of rooted forests. That is, if P is the partial order formed by the reachability relationship between vertices in a rooted forest, then the comparability graph of P is trivially perfect, and every

364 100.2. RELATED CLASSES OF GRAPHS 365

trivially perfect graph can be formed in this way.[5]

[6] • They are the graphs that do not have a P4 path graph or a C4 cycle graph as induced subgraphs. • They are the graphs that can be represented as the interval graphs for a set of nested intervals. A set of intervals is nested if, for every two intervals in the set, either the two are disjoint or one contains the other.[7] • They are the graphs that are both chordal and cographs.[8] This follows from the characterization of chordal graphs as the graphs without induced cycles of length greater than three, and of cographs as the graphs without induced paths on four vertices (P4). • They are the graphs that are both cographs and interval graphs.[8] • They are the graphs that can be formed, starting from one-vertex graphs, by two operations: disjoint union of two smaller trivially perfect graphs, and the addition of a new vertex adjacent to all the vertices of a smaller trivially perfect graph.[9] These operations correspond, in the underlying forest, to forming a new forest by the disjoint union of two smaller forests and forming a tree by connecting a new root node to the roots of all the trees in a forest. • They are the graphs in which, for every edge uv, the neighborhoods of u and v (including u and v themselves) are nested: one neighborhood must be a subset of the other.[10] • They are the permutation graphs deﬁned from stack-sortable permutations.[11] • They are the graphs with the property that in each of its induced subgraphs the clique cover number equals the number of maximal cliques.[12] • They are the graphs with the property that in each of its induced subgraphs the clique number equals the pseudo-Grundy number.[12] • They are the graphs with the property that in each of its induced subgraphs the chromatic number equals the pseudo-Grundy number.[12]

100.2 Related classes of graphs

It follows from the equivalent characterizations of trivially perfect graphs that every trivially perfect graph is also a cograph, a chordal graph, an interval graph, and a perfect graph. The threshold graphs are exactly the graphs that are both themselves trivially perfect and the complements of trivially perfect graphs (co-trivially perfect graphs).[13] Windmill graphs are trivially perfect.

100.3 Recognition

Chu (2008) describes a simple linear time algorithm for recognizing trivially perfect graphs, based on lexicographic breadth-first search. Whenever the LexBFS algorithm removes a vertex v from the first set on its queue, the algorithm checks that all remaining neighbors of v belong to the same set; if not, one of the forbidden induced subgraphs can be constructed from v. If this check succeeds for every v, then the graph is trivially perfect. The algorithm can also be modified to test whether a graph is the complement graph of a trivially perfect graph, in linear time.

100.4 Notes

[1] Brandstädt, Le & Spinrad (1999), deﬁnition 2.6.2, p.34; Golumbic (1978).

[2] Wolk (1962); Wolk (1965).

[3] Donnelly & Isaak (1999).

[4] Yan, Chen & Chang (1996). 366 CHAPTER 100. TRIVIALLY PERFECT GRAPH

[5] Brandstädt, Le & Spinrad (1999), theorem 6.6.1, p. 99; Golumbic (1978), corollary 4.

[6] Brandstädt, Le & Spinrad (1999), theorem 6.6.1, p. 99; Golumbic (1978), theorem 2. Wolk (1962) and Wolk (1965) proved this for comparability graphs of rooted forests.

[7] Brandstädt, Le & Spinrad (1999), p. 51.

[8] Brandstädt, Le & Spinrad (1999), p. 248; Yan, Chen & Chang (1996), theorem 3.

[9] Yan, Chen & Chang (1996); Gurski (2006).

[10] Yan, Chen & Chang (1996), theorem 3.

[11] Rotem (1981).

[12] Rubio-Montiel (2015).

[13] Brandstädt, Le & Spinrad (1999), theorem 6.6.3, p. 100; Golumbic (1978), corollary 5.

100.5 References

• Brandstädt, Andreas; Le, Van Bang; Spinrad, Jeremy (1999), Graph Classes: A Survey, SIAM Monographs on Discrete Mathematics and Applications, ISBN 0-89871-432-X. • Chu, Frank Pok Man (2008), “A simple linear time certifying LBFS-based algorithm for recognizing trivially perfect graphs and their complements”, Information Processing Letters 107 (1): 7–12, doi:10.1016/j.ipl.2007.12.009. • Donnelly, Sam; Isaak, Garth (1999), “Hamiltonian powers in threshold and arborescent comparability graphs”, Discrete Mathematics 202 (1-3): 33–44, doi:10.1016/S0012-365X(98)00346-X • Golumbic, Martin Charles (1978), “Trivially perfect graphs”, Discrete Mathematics 24 (1): 105–107, doi:10.1016/0012- 365X(78)90178-4. • Gurski, Frank (2006), “Characterizations for co-graphs deﬁned by restricted NLC-width or clique-width operations”, Discrete Mathematics 306 (2): 271–277, doi:10.1016/j.disc.2005.11.014. • Rotem, D. (1981), “Stack sortable permutations”, Discrete Mathematics 33 (2): 185–196, doi:10.1016/0012- 365X(81)90165-5, MR 599081. • Rubio-Montiel, C. (2015), “A new characterization of trivially perfect graphs”, Electronic Journal of Graph Theory and Applications 3 (1): 22–26, doi:10.5614/ejgta.2015.3.1.3. • Wolk, E. S. (1962), “The comparability graph of a tree”, Proceedings of the American Mathematical Society (5 ed.) 13: 789–795, doi:10.1090/S0002-9939-1962-0172273-0. • Wolk, E. S. (1965), “A note on the comparability graph of a tree”, Proceedings of the American Mathematical Society (1 ed.) 16: 17–20, doi:10.1090/S0002-9939-1965-0172274-5. • Yan, Jing-Ho; Chen, Jer-Jeong; Chang, Gerard J. (1996), “Quasi-threshold graphs”, Discrete Applied Mathe- matics 69 (3): 247–255, doi:10.1016/0166-218X(96)00094-7.

100.6 External links

• “Trivially perfect graphs”, Information System on Graph Classes and their Inclusions Chapter 101

Universal graph

In mathematics, a universal graph is an infinite graph that contains every finite (or at-most-countable) graph as an induced subgraph. A universal graph of this type was first constructed by R. Rado[1][2] and is now called the Rado graph or random graph. More recent work[3] [4] has focused on universal graphs for a graph family F: that is, an infinite graph belonging to F that contains all finite graphs in F. For instance, the Henson graphs are universal in this sense for the i-clique-free graphs. A universal graph for a family F of graphs can also refer to a member of a sequence of finite graphs that contains all graphs in F; for instance, every finite tree is a subgraph of a sufficiently large hypercube graph[5] so a hypercube can be said to be a universal graph for trees. However it is not the smallest such graph: it is known that there is a universal graph for n-node trees with only n vertices and O(n log n) edges, and that this is optimal.[6] A construction based on the planar separator theorem can be used to show that n-vertex planar graphs have universal graphs with O(n3/2) edges, and that bounded-degree planar graphs have universal graphs with O(n log n) edges.[7][8][9] Sumner’s conjecture states that tournaments are universal for polytrees, in the sense that every tournament with 2n − 2 vertices contains every polytree with n vertices as a subgraph.[10] A family F of graphs has a universal graph of polynomial size, containing every n-vertex graph as an induced subgraph, if and only if it has an adjacency labelling scheme in which vertices may be labeled by O(log n)-bit bitstrings such that an algorithm can determine whether two vertices are adjacent by examining their labels. For, if a universal graph of this type exists, the vertices of any graph in F may be labeled by the identities of the corresponding vertices in the universal graph, and conversely if a labeling scheme exists then a universal graph may be constructed having a vertex for every possible label.[11] In older mathematical terminology, the phrase “universal graph” was sometimes used to denote a complete graph.

101.1 References

[1] Rado, R. (1964). “Universal graphs and universal functions”. Acta Arithmetica 9: 331–340. MR 0172268.

[2] Rado, R. (1967). “Universal graphs”. A Seminar in Graph Theory. Holt, Rinehart, and Winston. pp. 83–85. MR 0214507.

[3] Goldstern, Martin; Kojman, Menachem (1996). “Universal arrow-free graphs”. Acta Mathematica Hungarica 1973 (4): 319–326. arXiv:math.LO/9409206. doi:10.1007/BF00052907. MR 1428038.

[4] Cherlin, Greg; Shelah, Saharon; Shi, Niandong (1999). “Universal graphs with forbidden subgraphs and algebraic closure”. Advances in Applied Mathematics 22 (4): 454–491. arXiv:math.LO/9809202. doi:10.1006/aama.1998.0641. MR 1683298.

[5] Wu, A. Y. (1985). “Embedding of tree networks into hypercubes”. Journal of Parallel and Distributed Computing 2 (3): 238–249. doi:10.1016/0743-7315(85)90026-7.

[6] Chung, F. R. K.; Graham, R. L. (1983). “On universal graphs for spanning trees” (PDF). Journal of the London Mathe- matical Society 27 (2): 203–211. doi:10.1112/jlms/s2-27.2.203. MR 0692525..

[7] Babai, L.; Chung, F. R. K.; Erdős, P.; Graham, R. L.; Spencer, J. H. (1982). “On graphs which contain all sparse graphs”. In Rosa, Alexander; Sabidussi, Gert; Turgeon, Jean. Theory and practice of combinatorics: a collection of articles honoring Anton Kotzig on the occasion of his sixtieth birthday (PDF). Annals of Discrete Mathematics 12. pp. 21–26. MR 0806964.

367 368 CHAPTER 101. UNIVERSAL GRAPH

[8] Bhatt, Sandeep N.; Chung, Fan R. K.; Leighton, F. T.; Rosenberg, Arnold L. (1989). “Universal graphs for bounded-degree trees and planar graphs”. SIAM Journal on Discrete Mathematics 2 (2): 145. doi:10.1137/0402014. MR 0990447.

[9] Chung, Fan R. K. (1990). “Separator theorems and their applications”. In Korte, Bernhard; Lovász, László; Prömel, Hans Jürgen; et al. Paths, Flows, and VLSI-Layout. Algorithms and Combinatorics 9. Springer-Verlag. pp. 17–34. ISBN 978-0-387-52685-0. MR 1083375.

[10] Sumner’s Universal Tournament Conjecture, Douglas B. West, retrieved 2010-09-17.

[11] Kannan, Sampath; Naor, Moni; Rudich, Steven (1992), “Implicit representation of graphs”, SIAM Journal on Discrete Mathematics 5 (4): 596–603, doi:10.1137/0405049, MR 1186827.

101.2 External links

• The panarborial formula, “Theorem of the Day” concerning universal graphs for trees Chapter 102

Vertex-transitive graph

In the mathematical ﬁeld of graph theory, a vertex-transitive graph is a graph G such that, given any two vertices v1 and v2 of G, there is some automorphism f : V (G) → V (G) such that

f(v1) = v2.

In other words, a graph is vertex-transitive if its automorphism group acts transitively upon its vertices.[1] A graph is vertex-transitive if and only if its graph complement is, since the group actions are identical. Every symmetric graph without isolated vertices is vertex-transitive, and every vertex-transitive graph is regular. However, not all vertex-transitive graphs are symmetric (for example, the edges of the truncated tetrahedron), and not all regular graphs are vertex-transitive (for example, the Frucht graph and Tietze’s graph).

102.1 Finite examples

Finite vertex-transitive graphs include the symmetric graphs (such as the Petersen graph, the Heawood graph and the vertices and edges of the Platonic solids). The ﬁnite Cayley graphs (such as cube-connected cycles) are also vertex-transitive, as are the vertices and edges of the Archimedean solids (though only two of these are symmetric). Potočnik, Spiga and Verret have constructed a census of all connected cubic vertex-transitive graphs on at most 1280 vertices.[2] Although every Cayley graph is vertex-transitive, there exist other vertex-transitive graphs that are not Cayley graphs. The most famous example is the Petersen graph, but others can be constructed including the line graphs of edge- transitive non-bipartite graphs with odd vertex degrees.[3]

102.2 Properties

The edge-connectivity of a vertex-transitive graph is equal to the degree d, while the vertex-connectivity will be at least 2(d+1)/3.[4] If the degree is 4 or less, or the graph is also edge-transitive, or the graph is a minimal Cayley graph, then the vertex-connectivity will also be equal to d.[5]

102.3 Inﬁnite examples

Inﬁnite vertex-transitive graphs include:

369 370 CHAPTER 102. VERTEX-TRANSITIVE GRAPH

The edges of the truncated tetrahedron form a vertex-transitive graph (also a Cayley graph) which is not symmetric.

• infinite paths (infinite in both directions) • infinite regular trees, e.g. the Cayley graph of the free group • graphs of uniform tessellations (see a complete list of planar tessellations), including all tilings by regular polygons • infinite Cayley graphs • the Rado graph

Two countable vertex-transitive graphs are called quasi-isometric if the ratio of their distance functions is bounded from below and from above. A well known conjecture stated that every inﬁnite vertex-transitive graph is quasi- isometric to a Cayley graph. A counterexample was proposed by Diestel and Leader in 2001.[6] In 2005, Eskin, Fisher, and Whyte conﬁrmed the counterexample.[7]

102.4 See also

• Edge-transitive graph • Lovász conjecture • Semi-symmetric graph 102.5. REFERENCES 371

• Zero-symmetric graph

102.5 References

[1] Godsil, Chris; Royle, Gordon (2001), Algebraic Graph Theory, Graduate Texts in Mathematics 207, New York: Springer- Verlag.

[2] Potočnik P., Spiga P. and Verret G. (2013), “Cubic vertex-transitive graphs on up to 1280 vertices”, Journal of Symbolic Computation 50: 465–477, doi:10.1016/j.jsc.2012.09.002.

[3] Lauri, Josef; Scapellato, Raﬀaele (2003), Topics in graph automorphisms and reconstruction, London Mathematical So- ciety Student Texts 54, Cambridge: Cambridge University Press, p. 44, ISBN 0-521-82151-7, MR 1971819. Lauri and Scapelleto credit this construction to Mark Watkins.

[4] Godsil, C. and Royle, G. (2001), Algebraic Graph Theory, Springer Verlag

[5] Babai, L. (1996), Technical Report TR-94-10, University of Chicago

[6] Diestel, Reinhard; Leader, Imre (2001), “A conjecture concerning a limit of non-Cayley graphs” (PDF), Journal of Alge- braic Combinatorics 14 (1): 17–25, doi:10.1023/A:1011257718029.

[7] Eskin, Alex; Fisher, David; Whyte, Kevin (2005). “Quasi-isometries and rigidity of solvable groups”. arXiv:math.GR/ 0511647..

102.6 External links

• Weisstein, Eric W., “Vertex-transitive graph”, MathWorld. • A census of small connected cubic vertex-transitive graphs . Primož Potočnik, Pablo Spiga, Gabriel Verret, 2012. Chapter 103

Well-covered graph

A well-covered graph, the intersection graph of the nine diagonals of a hexagon. The three red vertices form one of its 14 equal-sized maximal independent sets, and the six blue vertices form the complementary minimal vertex cover.

In graph theory, a well-covered graph is an undirected graph in which every minimal vertex cover has the same size as every other minimal vertex cover. Well-covered graphs were deﬁned and ﬁrst studied by Plummer (1970).

372 103.1. DEFINITIONS 373

103.1 Deﬁnitions

A vertex cover in a graph is a set of vertices that touches every edge in the graph. A vertex cover is minimal, or irredundant, if removing any vertex from it would destroy the covering property. It is minimum if there is no other vertex cover with fewer vertices. A well-covered graph is one in which every minimal cover is also minimum; in the original paper defining well-covered graphs, Plummer (1970) writes that this is “obviously equivalent” to the property that every two minimal covers have the same number of vertices as each other. An independent set in a graph is a set of vertices no two of which are adjacent to each other. If C is a vertex cover in a graph G, the complement of C must be an independent set, and vice versa. C is a minimal vertex cover if and only if its complement I is a maximal independent set, and C is a minimum vertex cover if and only if its complement is a maximum independent set. Therefore, a well-covered graph is, equivalently, a graph in which every maximal independent set has the same size, or a graph in which every maximal independent set is maximum. In the original paper defining well-covered graphs, these definitions were restricted to apply only to connected graphs,[1] although they are meaningful for disconnected graphs as well. Some later authors have replaced the connectivity requirement with the weaker requirement that a well-covered graph must not have any isolated vertices.[2] For both connected well-covered graphs and well-covered graphs without isolated vertices, there can be no essential vertices, vertices which belong to every minimum vertex cover.[1] Additionally, every well-covered graph is a critical graph for vertex covering in the sense that, for every vertex v, deleting v from the graph produces a graph with a smaller minimum vertex cover.[1] The independence complex of a graph G is the simplicial complex having a simplex for each independent set in G. A simplicial complex is said to be pure if all its maximal simplices have the same cardinality, so a well-covered graph is equivalently a graph with a pure independence complex.[3]

103.2 Examples

A non-attacking placement of eight rooks on a chessboard. If fewer than eight rooks are placed in a non-attacking way on a chessboard, they can always be extended to eight rooks that remain non-attacking.

A cycle graph of length four or ﬁve is well-covered: in each case, every maximal independent set has size two. A cycle of length seven, and a path of length three, are also well-covered. Every complete graph is well-covered: every maximal independent set consists of a single vertex. A complete bipartite graph is well covered if the two sides of its bipartition have equal numbers of vertices, for these are its only two maximal independent sets. A rook’s graph is well covered: if one places any set of rooks on a chessboard so that no two rooks are attacking each other, it is always possible to continue placing more non-attacking rooks until there is one on every row and column of the chessboard. If P is either a polygon or a set of points, S is the set of open line segments that have vertices of P as endpoints and are otherwise disjoint from P, and G is the intersection graph of S (a graph that has a vertex for each line segment in S and an edge for each two line segments that cross each other), then G is well-covered. For in this case, every maximal independent set in G corresponds to the set of edges in a triangulation of P, and a calculation involving the Euler characteristic shows that every two triangulations have the same number of edges as each other.[4] If G is any n-vertex graph, then the rooted product of G with a one-edge graph (that is, the graph H formed by adding n new vertices to G, each of degree one and each adjacent to a distinct vertex in G) is well-covered. For, a maximal independent set in H must consist of an arbitrary independent set in G together with the degree-one neighbors of the complementary set, and must therefore have size n.[5] More generally, given any graph G together with a clique cover (a partition p of the vertices of G into cliques), the graph Gp formed by adding another vertex to each clique is well-covered; the rooted product is the special case of this construction when p consists of n one-vertex cliques.[6] Thus, every graph is an induced subgraph of a well-covered graph.

103.3 Bipartiteness, very well covered graphs, and girth

Favaron (1982) deﬁnes a very well covered graph to be a well-covered graph (possibly disconnected, but with no isolated vertices) in which each maximal independent set (and therefore also each minimal vertex cover) contains exactly half of the vertices. This deﬁnition includes the rooted products of a graph G and a one-edge graph. It also 374 CHAPTER 103. WELL-COVERED GRAPH includes, for instance, the bipartite well-covered graphs studied by Ravindra (1977) and Berge (1981): in bipartite graph without isolated vertices, both sides of any bipartition form maximal independent sets (and minimal vertex covers), so if the graph is well-covered both sides must have equally many vertices. In a well-covered graph with n vertices, the size of a maximum independent set is at most n/2, so very well covered graphs are the well covered graphs in which the maximum independent set size is as large as possible.[7] A bipartite graph G is well-covered if and only if it has a perfect matching M with the property that, for every edge uv in M, the induced subgraph of the neighbors of u and v forms a complete bipartite graph.[8] The characterization in terms of matchings can be extended from bipartite graphs to very well covered graphs: a graph G is very well covered if and only if it has a perfect matching M with the following two properties:

1. No edge of M belongs to a triangle in G, and

2. If an edge of M is the central edge of a three-edge path in G, then the two endpoints of the path must be adjacent.

Moreover, if G is very well covered, then every perfect matching in G satisﬁes these properties.[9] Trees are a special case of bipartite graphs, and testing whether a tree is well-covered can be handled as a much simpler special case of the characterization of well-covered bipartite graphs: if G is a tree with more than two vertices, it is well-covered if and only if each non-leaf node of the tree is adjacent to exactly one leaf.[8] The same characterization applies to graphs that are locally tree-like, in the sense that low-diameter neighborhoods of every vertex are acyclic: if a graph has girth eight or more (so that, for every vertex v, the subgraph of vertices within distance three of v is acyclic) then it is well-covered if and only if every vertex of degree greater than one has exactly one neighbor of degree one.[10] A closely related but more complex characterization applies to well-covered graphs of girth ﬁve or more.[11]

103.4 Regularity and planarity

The seven cubic 3-connected well-covered graphs

The cubic (3-regular) well-covered graphs have been classiﬁed: they consist of seven 3-connected examples, together with three inﬁnite families of cubic graphs with lesser connectivity.

The seven 3-connected cubic well-covered graphs are the complete graph K4, the graphs of the triangular prism and the pentagonal prism, the Dürer graph, the utility graph K₃,₃, an eight-vertex graph obtained from the utility graph by a Y-Δ transform, and the 14-vertex generalized Petersen graph G(7,2).[12] Of these graphs, the ﬁrst four are planar graphs and therefore also the only four cubic polyhedral graphs (graphs of simple convex polyhedra) that are well-covered. Four of the graphs (the two prisms, the Dürer graph, and G(7,2)) are generalized Petersen graphs. 103.5. COMPLEXITY 375

CB A

CA B

A cubic 1-connected well-covered graph, formed by replacing each node of a six-node path by one of three fragments

The 1- and 2-connected cubic well-covered graphs are all formed by replacing the nodes of a path or cycle by three fragments of graphs which Plummer (1993) labels A, B, and C. Fragments A or B may be used to replace the nodes of a cycle or the interior nodes of a path, while fragment C is used to replace the two end nodes of a path. Fragment A contains a bridge, so the result of performing this replacement process on a path and using fragment A to replace some of the path nodes (and the other two fragments for the remaining nodes) is a 1-vertex-connected cubic well-covered graph. All 1-vertex-connected cubic well-covered graphs have this form, and all such graphs are planar.[13] There are two types of 2-vertex-connected cubic well-covered graphs. One of these two families is formed by replacing the nodes of a cycle by fragments A and B, with at least two of the fragments being of type A; a graph of this type is planar if and only if it does not contain any fragments of type B. The other family is formed by replacing the nodes of a path by fragments of type B and C; all such graphs are planar.[13] Complementing the characterization of well-covered simple polyhedra in three dimensions, researchers have also considered the well-covered simplicial polyhedra, or equivalently the well-covered maximal planar graphs. Every maximal planar graph with ﬁve or more vertices has vertex connectivity 3, 4, or 5.[14] There are no well-covered 5-connected maximal planar graphs, and there are only four 4-connected well-covered maximal planar graphs: the graphs of the regular octahedron, the pentagonal dipyramid, the snub disphenoid, and an irregular polyhedron (a nonconvex deltahedron) with 12 vertices, 30 edges, and 20 triangular faces. However, there are inﬁnitely many 3- connected well-covered maximal planar graphs.[15] For instance, a well-covered 3-connected maximal planar graph may be obtained via the clique cover construction[6] from any 3t-vertex maximal planar graph in which there are t disjoint triangle faces by adding t new vertices, one within each of these faces.

103.5 Complexity

Testing whether a graph contains two maximal independent sets of different sizes is NP-complete; that is, com- plementarily, testing whether a graph is well-covered is coNP-complete.[16] Although it is easy to find maximum independent sets in graphs that are known to be well-covered, it is also NP-hard for an algorithm to produce as output, on all graphs, either a maximum independent set or a guarantee that the input is not well-covered.[17] In contrast, it is possible to test whether a given graph G is very well covered in polynomial time. To do so, find the subgraph H of G consisting of the edges that satisfy the two properties of a matched edge in a very well covered graph, and then use a matching algorithm to test whether H has a perfect matching.[9] Some problems that are NP-complete for arbitrary graphs, such as the problem of finding a Hamiltonian cycle, may also be solved in polynomial time for 376 CHAPTER 103. WELL-COVERED GRAPH

The snub disphenoid, one of four well-covered 4-connected 3-dimensional simplicial polyhedra. very well covered graphs.[18] A graph is said to be equimatchable if every maximal matching is maximum; that is, it is equimatchable if its line graph is well-covered. It is possible to test whether a line graph, or more generally a claw-free graph, is well-covered in polynomial time.[19] The characterizations of well-covered graphs with girth ﬁve or more, and of well-covered graphs that are 3-regular, also lead to eﬃcient polynomial time algorithms to recognize these graphs.[20]

103.6 Notes

[1] Plummer (1970).

[2] Favaron (1982).

[3] For examples of papers using this terminology, see Dochtermann & Engström (2009) and Cook & Nagel (2010).

[4] Greenberg (1993).

[5] This class of examples was studied by Fink et al. (1985); they are also (together with the four-edge cycle, which is also 103.7. REFERENCES 377

well-covered) exactly the graphs whose domination number is n/2. Its well-covering property is also stated in diﬀerent terminology (having a pure independence complex) as Theorem 4.4 of Dochtermann & Engström (2009).

[6] Cook & Nagel (2010).

[7] Berge (1981).

[8] Ravindra (1977); Plummer (1993).

[9] Staples (1975); Favaron (1982); Plummer (1993).

[10] Finbow & Hartnell (1983); Plummer (1993), Theorem 4.1.

[11] Finbow & Hartnell (1983); Plummer (1993), Theorem 4.2.

[12] Campbell (1987); Finbow, Hartnell & Nowakowski (1988); Campbell, Ellingham & Royle (1993); Plummer (1993).

[13] Campbell (1987); Campbell & Plummer (1988); Plummer (1993).

[14] The complete graphs on 1, 2, 3, and 4 vertices are all maximal planar and well-covered; their vertex connectivity is either unbounded or at most three, depending on details of the deﬁnition of vertex connectivity that are irrelevant for larger maximal planar graphs.

[15] Finbow, Hartnell, and Nowakowski et al. (2003, 2009, 2010).

[16] Sankaranarayana & Stewart (1992); Chvátal & Slater (1993); Caro, Sebő & Tarsi (1996).

[17] Raghavan & Spinrad (2003).

[18] Sankaranarayana & Stewart (1992).

[19] Lesk, Plummer & Pulleyblank (1984); Tankus & Tarsi (1996); Tankus & Tarsi (1997).

[20] Campbell, Ellingham & Royle (1993); Plummer (1993).

103.7 References

• Berge, Claude (1981), “Some common properties for regularizable graphs, edge-critical graphs and B-graphs”, Graph theory and algorithms (Proc. Sympos., Res. Inst. Electr. Comm., Tohoku Univ., Sendai, 1980), Lec- ture Notes in Computer Science 108, Berlin: Springer, pp. 108–123, doi:10.1007/3-540-10704-5_10, MR 622929. • Campbell, S. R. (1987), Some results on cubic well-covered graphs, Ph.D. thesis, Vanderbilt University, De- partment of Mathematics. As cited by Plummer (1993). • Campbell, S. R.; Ellingham, M. N.; Royle, Gordon F. (1993), “A characterisation of well-covered cubic graphs”, Journal of Combinatorial Mathematics and Combinatorial Computing 13: 193–212, MR 1220613. • Campbell, Stephen R.; Plummer, Michael D. (1988), “On well-covered 3-polytopes”, Ars Combinatoria 25 (A): 215–242, MR 942505. • Caro, Yair; Sebő, András; Tarsi, Michael (1996), “Recognizing greedy structures”, Journal of Algorithms 20 (1): 137–156, doi:10.1006/jagm.1996.0006, MR 1368720. • Chvátal, Václav; Slater, Peter J. (1993), “A note on well-covered graphs”, Quo vadis, graph theory?, Annals of Discrete Mathematics 55, Amsterdam: North-Holland, pp. 179–181, MR 1217991. • Cook, David, II; Nagel, Uwe (2010), Cohen-Macaulay graphs and face vectors of ﬂag complexes, arXiv:1003.4447. • Dochtermann, Anton; Engström, Alexander (2009), “Algebraic properties of edge ideals via combinatorial topology”, Electronic Journal of Combinatorics 16 (2): Research Paper 2, MR 2515765. • Favaron, O. (1982), “Very well covered graphs”, Discrete Mathematics 42 (2-3): 177–187, doi:10.1016/0012- 365X(82)90215-1, MR 677051. • Finbow, A. S.; Hartnell, B. L. (1983), “A game related to covering by stars”, Ars Combinatoria 16 (A): 189– 198, MR 737090. 378 CHAPTER 103. WELL-COVERED GRAPH

• Finbow, A.; Hartnell, B.; Nowakowski, R. (1988), “Well-dominated graphs: a collection of well-covered ones”, Ars Combinatoria 25 (A): 5–10, MR 942485. • Finbow, A.; Hartnell, B.; Nowakowski, R. J. (1993), “A characterization of well covered graphs of girth 5 or greater”, Journal of Combinatorial Theory, Series B 57 (1): 44–68, doi:10.1006/jctb.1993.1005, MR 1198396. • Finbow, A.; Hartnell, B.; Nowakowski, R.; Plummer, Michael D. (2003), “On well-covered triangulations. I”, Discrete Applied Mathematics 132 (1-3): 97–108, doi:10.1016/S0166-218X(03)00393-7, MR 2024267. • Finbow, Arthur S.; Hartnell, Bert L.; Nowakowski, Richard J.; Plummer, Michael D. (2009), “On well-covered triangulations. II”, Discrete Applied Mathematics 157 (13): 2799–2817, doi:10.1016/j.dam.2009.03.014, MR 2537505.

• Finbow, Arthur S.; Hartnell, Bert L.; Nowakowski, Richard J.; Plummer, Michael D. (2010), “On well-covered triangulations. III”, Discrete Applied Mathematics 158 (8): 894–912, doi:10.1016/j.dam.2009.08.002, MR 2602814.

• Fink, J. F.; Jacobson, M. S.; Kinch, L. F.; Roberts, J. (1985), “On graphs having domination number half their order”, Period. Math. Hungar. 16 (4): 287–293, doi:10.1007/BF01848079, MR 0833264.

• Greenberg, Peter (1993), “Piecewise SL2Z geometry”, Transactions of the American Mathematical Society 335 (2): 705–720, doi:10.2307/2154401, MR 1140914.

• Lesk, M.; Plummer, M. D.; Pulleyblank, W. R. (1984), “Equi-matchable graphs”, Graph theory and combinatorics (Cambridge, 1983), London: Academic Press, pp. 239–254, MR 777180.

• Plummer, Michael D. (1970), “Some covering concepts in graphs”, Journal of Combinatorial Theory 8: 91–98, doi:10.1016/S0021-9800(70)80011-4, MR 0289347.

• Plummer, Michael D. (1993), “Well-covered graphs: a survey”, Quaestiones Mathematicae 16 (3): 253–287, doi:10.1080/16073606.1993.9631737, MR 1254158.

• Raghavan, Vijay; Spinrad, Jeremy (2003), “Robust algorithms for restricted domains”, Journal of Algorithms 48 (1): 160–172, doi:10.1016/S0196-6774(03)00048-8, MR 2006100.

• Ravindra, G. (1977), “Well-covered graphs”, Journal of Combinatorics, Information and System Sciences 2 (1): 20–21, MR 0469831.

• Sankaranarayana, Ramesh S.; Stewart, Lorna K. (1992), “Complexity results for well-covered graphs”, Net- works 22 (3): 247–262, doi:10.1002/net.3230220304, MR 1161178.

• Staples, J. (1975), On some subclasses of well-covered graphs, Ph.D. thesis, Vanderbilt University, Department of Mathematics. As cited by Plummer (1993).

• Tankus, David; Tarsi, Michael (1996), “Well-covered claw-free graphs”, Journal of Combinatorial Theory, Series B 66 (2): 293–302, doi:10.1006/jctb.1996.0022, MR 1376052.

• Tankus, David; Tarsi, Michael (1997), “The structure of well-covered graphs and the complexity of their recognition problems”, Journal of Combinatorial Theory, Series B 69 (2): 230–233, doi:10.1006/jctb.1996.1742, MR 1438624. Chapter 104

Zero-symmetric graph

The smallest zero-symmetric graph, with 18 vertices and 27 edges

The truncated cuboctahedron, a zero-symmetric polyhedron

In the mathematical ﬁeld of graph theory, a zero-symmetric graph is a connected graph in which all vertices are symmetric to each other, each vertex has exactly three incident edges, and these three edges are not symmetric to each other. More precisely, it is a connected vertex-transitive cubic graph whose edges are partitioned into three diﬀerent orbits by the automorphism group.[1] In these graphs, for every two vertices u and v, there is exactly one graph automorphism that takes u into v.[2] The name for this class of graphs was coined by R. M. Foster in a 1966 letter to H. S. M. Coxeter.[3]

104.1 Examples

The smallest zero-symmetric graph is a nonplanar graph with 18 vertices.[4] Its LCF notation is [5,−5]9.

379 380 CHAPTER 104. ZERO-SYMMETRIC GRAPH

Among planar graphs, the truncated cuboctahedral and truncated icosidodecahedral graphs are also zero-symmetric.[5] These examples are all bipartite graphs. However, there exist larger examples of zero-symmetric graphs that are not bipartite.[6]

104.2 Properties

Every finite zero-symmetric graph is a Cayley graph, a property that does not always hold for cubic vertex-transitive graphs more generally and that helps in the solution of combinatorial enumeration tasks concerning zero-symmetric graphs. There are 97687 zero-symmetric graphs on up to 1280 vertices. These graphs form 89% of the cubic Cayley graphs and 88% of all connected vertex-transitive cubic graphs on the same number of vertices.[7] All known finite connected zero-symmetric graphs contain a Hamiltonian cycle, but it is unknown whether every finite connected zero-symmetric graph is necessarily Hamiltonian.[8] This is a special case of the Lovász conjecture that (with five known exceptions, none of which is zero-symmetric) every finite connected vertex-transitive graph and every finite Cayley graph is Hamiltonian.

104.3 See also

• Semi-symmetric graph, graphs that have symmetries between every two edges but not between every two vertices (reversing the roles of edges and vertices in the deﬁnition of zero-symmetric graphs)

104.4 References

[1] Coxeter, Harold Scott MacDonald; Frucht, Roberto; Powers, David L. (1981), Zero-symmetric graphs, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, ISBN 0-12-194580-4, MR 658666

[2] Coxeter, Frucht & Powers (1981), p. 4.

[3] Coxeter, Frucht & Powers (1981), p. ix.

[4] Coxeter, Frucht & Powers (1981), Figure 1.1, p. 5.

[5] Coxeter, Frucht & Powers (1981), pp. 75 and 80.

[6] Coxeter, Frucht & Powers (1981), p. 55.

[7] Potočnik, Primož; Spiga, Pablo; Verret, Gabriel (2013), “Cubic vertex-transitive graphs on up to 1280 vertices”, Journal of Symbolic Computation 50: 465–477, arXiv:1201.5317, doi:10.1016/j.jsc.2012.09.002, MR 2996891.

[8] Coxeter, Frucht & Powers (1981), p. 10. 104.5. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 381

104.5 Text and image sources, contributors, and licenses

104.5.1 Text

• Antiprism graph Source: https://en.wikipedia.org/wiki/Antiprism_graph?oldid=702615144 Contributors: Michael Hardy, Tomruen, BD2412, Dr Greg and David Eppstein • Aperiodic graph Source: https://en.wikipedia.org/wiki/Aperiodic_graph?oldid=544554949 Contributors: Michael Hardy, Rbraunwa, Cobaltbluetony, Rjwilmsi, Quuxplusone, Novwik, Dl2000, DivideByZero14, David Eppstein, Addbot, Citation bot 1, Matt Kwan and Anonymous: 2 • Apex graph Source: https://en.wikipedia.org/wiki/Apex_graph?oldid=702553117 Contributors: Gaius Cornelius, Chris the speller, David Eppstein, R'n'B, Citation bot, Gilo1969, Khazar2, Johnroper100 and Anonymous: 1 • Apollonian network Source: https://en.wikipedia.org/wiki/Apollonian_network?oldid=702519502 Contributors: Rjwilmsi, Myasuda, Cydebot, David Eppstein, Niceguyedc, Helpful Pixie Bot, Ctsourak, Andreschulz, FritzFasbinder and Anonymous: 2 • Asymmetric graph Source: https://en.wikipedia.org/wiki/Asymmetric_graph?oldid=542839248 Contributors: Altenmann, Bkell, Head- bomb, David Eppstein, Radagast3, Addbot, Luckas-bot, EmausBot and Anonymous: 1 • Biased graph Source: https://en.wikipedia.org/wiki/Biased_graph?oldid=710620798 Contributors: Michael Hardy, Charles Matthews, Giftlite, ArnoldReinhold, Zaslav, BD2412, Josh Parris, Modify, Bluebot, David Eppstein, R'n'B, Luis Goddyn, BG19bot and Anonymous: 3 • Biclique-free graph Source: https://en.wikipedia.org/wiki/Biclique-free_graph?oldid=696517624 Contributors: Rjwilmsi, David Epp- stein and Anonymous: 1 • Biconnected graph Source: https://en.wikipedia.org/wiki/Biconnected_graph?oldid=702633088 Contributors: McKay, Giftlite, Kmote, Mboverload, BD2412, Maxal, Open2universe, Mikmorg, Cydebot, David Eppstein, Tayste, Addbot, WikitanvirBot, ZéroBot, Jeremy112233 and Anonymous: 3 • Bipartite graph Source: https://en.wikipedia.org/wiki/Bipartite_graph?oldid=711548431 Contributors: Jdpipe, Nonenmac, Michael Hardy, Booyabazooka, Manojmp, TakuyaMurata, Delirium, Eric119, Altenmann, Netpilot43556, MathMartin, Giftlite, Tom harrison, Tobo~enwiki, Corti, Shahab, Paul August, Rgdboer, Bobo192, Mdd, Jérôme, Burn, Bkkbrad, BD2412, Brighterorange, FlaBot, Kri, YurikBot, Freiberg, Catgofire, Jpbowen, Nethgirb, Bota47, H@r@ld, Melchoir, BiT, DHN-bot~enwiki, Tsca.bot, Gassa, Jon Awbrey, Michael Dinolfo, Ojan, CRGreathouse, Flamholz, Cydebot, Kozuch, Thijs!bot, Jdudar, MistWiz, AndreasWittenstein, Kevinmon, David Eppstein, JoergenB, Robin S, Hpfister, Robert Illes, Policron, AlnoktaBOT, PaulTanenbaum, Jason Klaus, Charliearcuri, Mild Bill Hiccup, Alexbot, Addbot, Luckas-bot, Yobot, Rubinbot, Twri, Miym, Shmomuffin, Locobot, Howard McCay, LucienBOT, D'ohBot, Doostdar- WKP, Pinethicket, RjwilmsiBot, Ripchip Bot, Ebrambot, Pgdx, ClueBot NG, Wcherowi, Hoorayforturtles, 149AFK, Nullzero, Helpful Pixie Bot, Solomon7968, AndiPersti, GeoffreyT2000, SofjaKovalevskaja and Anonymous: 66 • Bivariegated graph Source: https://en.wikipedia.org/wiki/Bivariegated_graph?oldid=696680432 Contributors: Michael Hardy, Giftlite, Andreas Kaufmann, D6, Igorpak, SmackBot, Mcld, David Eppstein, Koko90, Cyfal, Atlantagirl~enwiki, Tangi-tamma, JRN08, Citation bot, Kitresaiba, Status quo not acceptable, At-par, GeoffreyT2000 and Anonymous: 1 • Block graph Source: https://en.wikipedia.org/wiki/Block_graph?oldid=642528036 Contributors: Zaslav, Rjwilmsi, Headbomb, David Eppstein, Addedentry, Ashokkrm, Srijanrshetty and Anonymous: 2 • Bound graph Source: https://en.wikipedia.org/wiki/Bound_graph?oldid=702633138 Contributors: BD2412, David Eppstein, PaulTane- nbaum, DOI bot, Twri, GoingBatty and Mark viking • Cactus graph Source: https://en.wikipedia.org/wiki/Cactus_graph?oldid=711552897 Contributors: Booyabazooka, Altenmann, Zaslav, BD2412, Maxal, Toffile, EAderhold, Thijswijs, Headbomb, Hermel, Scanlan, Mrld, David Eppstein, Cuzkatzimhut, Justin W Smith, Addbot, Marine2323, Casmemsnano, Citation bot, Twri, Thore Husfeldt, Citation bot 1, RjwilmsiBot, TilonRespir, ZéroBot, FHannes, Solomon7968, Andreschulz, Teika kazura, BattyBot, Kub1x, Luisalfredomoctezuma and Anonymous: 4 • Cage (graph theory) Source: https://en.wikipedia.org/wiki/Cage_(graph_theory)?oldid=712902365 Contributors: XJaM, Michael Hardy, Booyabazooka, McKay, Giftlite, John Baez, Cmglee, Bird of paradox, Myasuda, Ntsimp, David Eppstein, Rocchini, Koko90, Tesquivello, Addbot, Citation bot, Twri, ArthurBot, XZeroBot, Citation bot 1, TobeBot, Bourdillon69, KLBot2, BG19bot, Tudor987 and Anonymous: 6 • Cayley graph Source: https://en.wikipedia.org/wiki/Cayley_graph?oldid=713515950 Contributors: Tomo, Michael Hardy, AugPi, Charles Matthews, Timwi, Dcoetzee, McKay, Sander123, Jaredwf, Pascalromon, Tobias Bergemann, Tosha, Giftlite, Dbenbenn, Tomruen, Tompw, Crisófilax, Quasicharacter, Shirimasen, PAR, Linas, BD2412, Rjwilmsi, Magidin, Mathbot, Rell Canis, Denisgomes, SmackBot, Mhss, RDBrown, Bazonka, Nbarth, Mhym, Digana, Jim.belk, Thijs!bot, Headbomb, David Eppstein, JoergenB, Robert Illes, TXiKiBoT, Arcfrk, Radagast3, Thehotelambush, JackSchmidt, Timhoooey, Watchduck, SockPuppetForTomruen, Addbot, Calculuslover, Leycec, Luckas-bot, Yobot, Rubinbot, GaborPete, Twri, Xqbot, BenzolBot, Andrew.clemens, HSNie, Fomalhautpi, EmausBot, R. J. Mathar, ClueBot NG, Wcherowi, Aeginn, BG19bot, Solomon7968, Suhagja, Azuredream89, WilliamJennings1989 and Anonymous: 39 • Chordal graph Source: https://en.wikipedia.org/wiki/Chordal_graph?oldid=708078773 Contributors: Michael Hardy, Altenmann, Bkell, Peter Kwok, Rich Farmbrough, Zaslav, Oliphaunt, Rjwilmsi, Tizio, Chobot, YurikBot, Michael Slone, Danielx, Michaelll, Lyonsam, JoeBot, Headbomb, Escarbot, AntiVandalBot, A3nm, David Eppstein, Ged.R, LokiClock, TXiKiBoT, Someguy1221, Jochgem, Dom- ination989, JP.Martin-Flatin, Alexbot, Addbot, DOI bot, Landon1980, Kilom691, Citation bot, ArthurBot, Citation bot 1, HRoestBot, EmausBot, Martygo, CountMacula, ClueBot NG, BG19bot, Tentinator, Lucasr.mal, AnWiPaFr and Anonymous: 26 • Circulant graph Source: https://en.wikipedia.org/wiki/Circulant_graph?oldid=710772279 Contributors: Zero0000, Giftlite, Maxal, Head- bomb, Stdazi, David Eppstein, JoergenB, Maproom, Twri, Dimpase, BG19bot, Mobrie06 and Anonymous: 1 • Claw-free graph Source: https://en.wikipedia.org/wiki/Claw-free_graph?oldid=710389986 Contributors: Michael Hardy, Paisa, Rich Farmbrough, Drange net, GregorB, Adking80, RDBury, Cdills, Alaibot, Headbomb, Vanish2, Tokenzero, David Eppstein, PaulTanen- baum, Robert Samal, Wpegden, Mr. Granger, Twri, Miym, Citation bot 1, PigFlu Oink, John of Reading and Anonymous: 5 382 CHAPTER 104. ZERO-SYMMETRIC GRAPH

• Cograph Source: https://en.wikipedia.org/wiki/Cograph?oldid=708056389 Contributors: Silverfish, Charles Matthews, Dcoetzee, Math- Martin, Paul August, Zaslav, Lectonar, Marianocecowski, LOL, Magister Mathematicae, BD2412, Mathbot, YurikBot, Davepape, Chris the speller, Taxipom, Headbomb, David Eppstein, PaulTanenbaum, Addbot, Yobot, Citation bot, Twri, FrescoBot, Citation bot 1, Mas- tergreg82, Faust17, BG19bot, AnWiPaFr and Anonymous: 8 • Comparability graph Source: https://en.wikipedia.org/wiki/Comparability_graph?oldid=706053077 Contributors: Michael Hardy, Gan- dalf61, Giftlite, Zaslav, Ott2, Eassin, David Eppstein, Gambette, PaulTanenbaum, Leen Droogendijk, Tangi-tamma, DOI bot, Yobot, Kilom691, Citation bot, Twri, Citation bot 1, Terrykel and Anonymous: 1 • Conference graph Source: https://en.wikipedia.org/wiki/Conference_graph?oldid=544584513 Contributors: Zaslav, Ntsimp, David Epp- stein, Addbot, Twri and Ricardo Ferreira de Oliveira • Convex bipartite graph Source: https://en.wikipedia.org/wiki/Convex_bipartite_graph?oldid=706052815 Contributors: Joriki, Ketil- trout, J. Finkelstein, David Eppstein, Justin W Smith, Citation bot 1, RjwilmsiBot, Helpful Pixie Bot and Monkbot • Critical graph Source: https://en.wikipedia.org/wiki/Critical_graph?oldid=702633468 Contributors: Michael Hardy, MathMartin, Peter Kwok, Stolee, BD2412, Adking80, Claygate, Bluebot, Stdazi, David Eppstein, Kope, Rocchini, Lwr314, Watchduck, AnomieBOT, GreyisthenewBlack, Citation bot 1, Anrnusna and Anonymous: 5 • Cubic graph Source: https://en.wikipedia.org/wiki/Cubic_graph?oldid=702633491 Contributors: Nonenmac, Edward, Booyabazooka, Ahoerstemeier, Silverfish, Charles Matthews, Dcoetzee, McKay, Altenmann, Kuszi, MathMartin, Henrygb, Giftlite, Dratman, Tomruen, CALR, Livajo, Arthena, BD2412, Rjwilmsi, NatusRoma, Chobot, Bluebot, Tsca.bot, Honnza, Ntsimp, Gfonsecabr, Headbomb, Stdazi, David Eppstein, R'n'B, Koko90, It Is Me Here, VolkovBot, Radagast3, Rybu, Watchduck, PixelBot, Philippe Giabbanelli, Addbot, Light- bot, Luckas-bot, Twri, Miym, Citation bot 1, H taammoli, Jonesey95, RjwilmsiBot, WikitanvirBot, ZéroBot, R. J. Mathar, WBritten and Anonymous: 15 • Dense graph Source: https://en.wikipedia.org/wiki/Dense_graph?oldid=709035915 Contributors: Kku, Jitse Niesen, Goochelaar, Lar- ryv, BD2412, Koavf, Ott2, Chortos-2, Taxipom, DanDanRevolution, Mwtoews, Headbomb, David Eppstein, ClueBot, Abrech, Addbot, Luckas-bot, Amirobot, Citation bot, Twri, Hoopje, Miym, MathsPoetry, JonDePlume, Citation bot 1, RjwilmsiBot, EmausBot, ZéroBot, Helpful Pixie Bot, Solomon7968, Ddmichael and Anonymous: 13 • Descartes snark Source: https://en.wikipedia.org/wiki/Descartes_snark?oldid=668482700 Contributors: Michael Hardy, Andreas Kauf- mann, Jimmy Pitt, FrescoBot, Gabriel.c.drummond.cole and Nathann.cohen • Disperser Source: https://en.wikipedia.org/wiki/Disperser?oldid=589387275 Contributors: LC~enwiki, Michael Hardy, Silverfish, Ma- trixFrog, Jaredwf, Giftlite, Msh210, Drbreznjev, Oleg Alexandrov, Mathbot, Kurykh, Optimale, Wizard191, Hermel, David Eppstein, Taketa, AnomieBOT, Materialscientist, Shadowjams, Jdhede, Ikaproduct, Liangren8 and Anonymous: 13 • Distance-hereditary graph Source: https://en.wikipedia.org/wiki/Distance-hereditary_graph?oldid=713552942 Contributors: Michael Hardy, Chris the speller, Lyonsam, Headbomb, David Eppstein, Bender2k14, Sun Creator, Yobot, Kilom691, Citation bot, Citation bot 1, ClueBot NG, Solomon7968, BattyBot, Hnridder and Anonymous: 5 • Distance-regular graph Source: https://en.wikipedia.org/wiki/Distance-regular_graph?oldid=708032686 Contributors: Altenmann, Giftlite, Zaslav, BD2412, GünniX, Bgwhite, RFBailey, Evilbu, Kier07, SmackBot, Jokes Free4Me, Ntsimp, David Eppstein, Snigbrook, MystBot, Addbot, Cooldev.iitkgp, Luckas-bot, Yobot, Twri, G.perarnau, John85, Kiefer.Wolfowitz, Nathann.cohen and Anonymous: 3 • Distance-transitive graph Source: https://en.wikipedia.org/wiki/Distance-transitive_graph?oldid=702633818 Contributors: Nonenmac, Michael Hardy, Ciphergoth, Jitse Niesen, Altenmann, Giftlite, Zaslav, Uffish, Stolee, BD2412, Dtrebbien, RFBailey, Melchoir, Ntsimp, Roice3, David Eppstein, Koko90, Maproom, PaulTanenbaum, Radagast3, Wpolly, Addbot, Luckas-bot, Kilom691, Citation bot, Twri, FrescoBot, G.perarnau, Citation bot 1, Jonesey95 and ZéroBot • Domination perfect graph Source: https://en.wikipedia.org/wiki/Dominating_set?oldid=704213350 Contributors: Michael Hardy, Charles Matthews, Igorpak, BD2412, Danielx, Evilbu, Benandorsqueaks, SmackBot, Ntsimp, Bodlaender, Headbomb, Stdazi, David Eppstein, SuneJ~enwiki, Dmcq, Justin W Smith, Addbot, DOI bot, Yobot, Citation bot, Xqbot, Miym, Thore Husfeldt, FrescoBot, Citation bot 1, PigFlu Oink, MastiBot, Gosha figosha, RjwilmsiBot, Emmagcohen, Isomorphismus, Wcherowi, Helpful Pixie Bot, Emerine, Infinitestory, Paragbansal, Klostermeyer and Anonymous: 18 • Dually chordal graph Source: https://en.wikipedia.org/wiki/Dually_chordal_graph?oldid=708935615 Contributors: Michael Hardy, Na- talya, Yobot, Frietjes, AnWiPaFr and Anonymous: 1 • Edge-transitive graph Source: https://en.wikipedia.org/wiki/Edge-transitive_graph?oldid=702633926 Contributors: Tomo, Michael Hardy, Silverfish, Jaredwf, Giftlite, RJFJR, BD2412, Eubot, Hairy Dude, Csab, Tamfang, Steelpillow, David Eppstein, Radagast3, Ad- dbot, Genusfour, EmausBot, ZéroBot, BattyBot and Anonymous: 4 • Even-hole-free graph Source: https://en.wikipedia.org/wiki/Even-hole-free_graph?oldid=683711116 Contributors: Igorpak, Lyonsam, David Eppstein, Ulric1313, Citation bot, Hnridder and Anonymous: 3 • Expander graph Source: https://en.wikipedia.org/wiki/Expander_graph?oldid=679377450 Contributors: LC~enwiki, Michael Hardy, Manojmp, Silverfish, Schneelocke, Charles Matthews, Reina riemann, Jaredwf, MathMartin, Merovingian, Nitishkorula, Giftlite, Tromer, Nomeata, PeterC, Vina, Bosmon, Andreas Kaufmann, Trevor MacInnis, Blokhead, Qutezuce, Kimbly, Nurban, Linas, Mathbot, Ewlya- hoocom, Sodin, Bgwhite, YurikBot, Dkostic, Guruparan18, Ott2, SmackBot, Bluebot, Kurykh, Mhym, Kevin Milans, Wizard191, Yl- loh, CRGreathouse, Rab V, Hermel, Magioladitis, Vanish2, Dxiao, Alienmercy, David Eppstein, Radagast3, SieBot, Taemyr, Shira kr, Legobot, Yobot, Zandr4, Citation bot, MathsPoetry, Nageh, Citation bot 1, RobinK, Gehilfen, Xnn, Leastsquare, WikitanvirBot, Kirela- gin, ChuispastonBot, Joel B. Lewis, Helpful Pixie Bot, Dpadgett, Harikine, Dexbot, Pchakka, Divyanshu.shende and Anonymous: 34 • Extractor (mathematics) Source: https://en.wikipedia.org/wiki/Extractor_(mathematics)?oldid=478974773 Contributors: LC~enwiki, Michael Hardy, Angela, MatrixFrog, Bloodshedder, Riddley, Jaredwf, Marc Venot, Tromer, Meswiss, ArnoldReinhold, Mani1, Oleg Alexandrov, Mathbot, Bluebot, Anuprao, Wizard191, David Eppstein, Gwern, Taemyr, Emeritusl, Ejosse1, Rcsprinter123 and Anony- mous: 6 • Factor-critical graph Source: https://en.wikipedia.org/wiki/Factor-critical_graph?oldid=702634183 Contributors: Michael Hardy, BD2412, Rjwilmsi, David Eppstein, R'n'B, Austinmohr, Justin W Smith, Twri, Citation bot 1, Helpful Pixie Bot and Anonymous: 2 104.5. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 383

• Forbidden graph characterization Source: https://en.wikipedia.org/wiki/Forbidden_graph_characterization?oldid=702518273 Con- tributors: Edemaine, Michael Hardy, Populus, Altenmann, Giftlite, Rjwilmsi, Tizio, Michael Slone, SmackBot, Headbomb, Mack2, David Eppstein, R'n'B, Koko90, Jdcrutch, PaulTanenbaum, Justin W Smith, Rohan4321, AnomieBOT, Citation bot, Twri, Gilo1969, Luis Goddyn, Citation bot 1, At-par, RobinK, Episcophagus, Helpful Pixie Bot, Solomon7968 and Anonymous: 6 • Gallery of named graphs Source: https://en.wikipedia.org/wiki/Gallery_of_named_graphs?oldid=682467923 Contributors: Zundark, Nonenmac, Michael Hardy, Booyabazooka, Taxman, McKay, Tomruen, Andreas Kaufmann, Brhaspati, Rjwilmsi, Koavf, Arbor, Nimur, Wavelength, SmackBot, Armend, Soumyasch, CmdrObot, Vyznev Xnebara, Ntsimp, Rocket000, King Bee, Headbomb, Cygri, T.Friedrich, David Eppstein, Mollwollfumble, Rocchini, Koko90, Maproom, Jochgem, Radagast3, Wpolly, Mrberryman, Auntof6, Addbot, The Bushranger, Yobot, Twri, FrescoBot, RjwilmsiBot, EmausBot, Dimpase, Nathann.cohen and Anonymous: 7 • Half-transitive graph Source: https://en.wikipedia.org/wiki/Half-transitive_graph?oldid=702636890 Contributors: BD2412, Myasuda, Radagast3, Addbot, Luckas-bot, Twri, Citation bot 1 and Helpful Pixie Bot • Halin graph Source: https://en.wikipedia.org/wiki/Halin_graph?oldid=695748383 Contributors: Michael Hardy, Giftlite, Rich Farm- brough, Rjwilmsi, Evilbu, Stdazi, David Eppstein, Addbot, Yobot, Twri, MathsPoetry, D.Lazard, RockMagnetist, Helpful Pixie Bot, Merinjose and Anonymous: 3 • Hanan grid Source: https://en.wikipedia.org/wiki/Hanan_grid?oldid=614541257 Contributors: Michael Hardy, Tobias Bergemann, Jef- frey Sharkey, Alksub, David Eppstein, Twri, Omnipaedista, Eda eng and Anonymous: 1 • Highly irregular graph Source: https://en.wikipedia.org/wiki/Highly_irregular_graph?oldid=702637052 Contributors: Michael Hardy, Stemonitis, BD2412, David Eppstein, Dthomsen8, MurphEngineer, HandThumb and Anonymous: 2 • Homogeneous graph Source: https://en.wikipedia.org/wiki/Homogeneous_graph?oldid=702637305 Contributors: Michael Hardy, BD2412, Malcolma, Racklever, David Eppstein and KoenDelaere • Hypohamiltonian graph Source: https://en.wikipedia.org/wiki/Hypohamiltonian_graph?oldid=696373422 Contributors: Michael Hardy, Booyabazooka, Igorpak, BD2412, Rjwilmsi, Stdazi, David Eppstein, Koko90, Addbot, Citation bot, Bean49, Miym, Citation bot 1, Trap- pist the monk, Josve05a, Staszek Lem, M.jooyandeh and Anonymous: 2 • Implication graph Source: https://en.wikipedia.org/wiki/Implication_graph?oldid=710968901 Contributors: Altenmann, Vadmium, GregorB, CBM, Thisisraja, David Eppstein, DavidCBryant, DOI bot, Twri, ClueBot NG, BG19bot, 0a.io and Anonymous: 2 • Integral graph Source: https://en.wikipedia.org/wiki/Integral_graph?oldid=542629668 Contributors: Michael Hardy, David Eppstein, Koko90, Addbot, Luckas-bot, Xqbot, Louperibot and Anonymous: 1 • K-edge-connected graph Source: https://en.wikipedia.org/wiki/K-edge-connected_graph?oldid=702637496 Contributors: Booyabazooka, BD2412, Rjwilmsi, KD5TVI, Mgccl, Mike Fikes, CBM, Flamholz, Sopoforic, Headbomb, David Eppstein, R'n'B, Radagast3, Justin W Smith, AmirOnWiki, WikHead, Addbot, Luckas-bot, Erik9bot, LucienBOT, Sjcjoosten, ZéroBot, Inozem, Larsborn, Dexbot and Anony- mous: 11 • K-tree Source: https://en.wikipedia.org/wiki/K-tree?oldid=702677966 Contributors: Ott2, Headbomb, Magioladitis, David Eppstein, R'n'B, AdrienChen, VolkovBot, EmausBot and Helpful Pixie Bot • K-vertex-connected graph Source: https://en.wikipedia.org/wiki/K-vertex-connected_graph?oldid=651779877 Contributors: Nonen- mac, Dcoetzee, McKay, Giftlite, Rjwilmsi, Mike Fikes, Sopoforic, Headbomb, Douglas R. White, A3nm, David Eppstein, Ratfox, TXiK- iBoT, Radagast3, Justin W Smith, DragonBot, Addbot, Luckas-bot, Citation bot, Miym, ZéroBot, Zynwyx, BG19bot and Anonymous: 11 • Laman graph Source: https://en.wikipedia.org/wiki/Laman_graph?oldid=699405382 Contributors: Charles Matthews, SmackBot, Jonny- mt, RobHar, David Eppstein, FisherQueen, Rigid~enwiki, Addbot, Citation bot, Twri, Citation bot 1, Victor Alexandrov, Trappist the monk, ChrisGualtieri and Anonymous: 1 • Lattice graph Source: https://en.wikipedia.org/wiki/Lattice_graph?oldid=711219687 Contributors: Altenmann, Giftlite, Dionyziz, BD2412, Lambiam, George100, Paxinum, David Eppstein, KylieTastic, MystBot, Addbot, Twri, MathsPoetry, Duoduoduo, ZéroBot, Cogiati, El Roih, Goldenarcher5 and Anonymous: 5 • Leaf power Source: https://en.wikipedia.org/wiki/Leaf_power?oldid=708520425 Contributors: Michael Hardy, David Eppstein and An- WiPaFr • Line graph Source: https://en.wikipedia.org/wiki/Line_graph?oldid=709595656 Contributors: Booyabazooka, Dominus, Dcoetzee, Dys- prosia, Jaredwf, Altenmann, MathMartin, Giftlite, Dbenbenn, DavidCary, Gscshoyru, Peter Kwok, Qef, Abdull, Discospinster, TedPavlic, Paul August, Zaslav, Danski14, Ahruman, Melaen, Oleg Alexandrov, Guardian of Light, Magister Mathematicae, Rjwilmsi, Vary, Vince Vatter, Chobot, YurikBot, Avraham, Reyk, Ilmari Karonen, Allens, DVD R W, SmackBot, Hydrogen Iodide, Gelingvistoj, Nbarth, Arch- dukefranz, Lyonsam, Ale jrb, CBM, Epbr123, Headbomb, AntiVandalBot, Braindrain0000, Oatmealcookiemon, Deadbeef, Hut 8.5, Dricherby, VoABot II, Animum, DropZone, David Eppstein, Seba5618, J.delanoy, SJP, PMajer, Onecanadasquarebishopsgate, Iain99, ClueBot, Excirial, Hayhay0147, Bender2k14, SoxBot III, Tangi-tamma, Addbot, Angrysockhop, Luckas-bot, Yobot, 2D, Ptbotgourou, Amirobot, Calle, AnomieBOT, Jim1138, Graham5571, Citation bot, OllieFury, Twri, LilHelpa, Capricorn42, Miym, Omnipaedista, Citation bot 1, Kitresaiba, Pinethicket, At-par, Meaghan, Merlion444, YouWillBeAssimilated, EmausBot, Lostinaforest, RA0808, John Cline, Josve05a, Confession0791, Arman Cagle, ClueBot NG, El Roih, Helpful Pixie Bot, Paolo Lipparini, Wiki13, Rutebega, Anbu121, Graphium, Anrnusna, Monkbot, Playerwithgraphs, Anon124 and Anonymous: 103 • Line graph of a hypergraph Source: https://en.wikipedia.org/wiki/Line_graph_of_a_hypergraph?oldid=702517375 Contributors: Michael Hardy, Jitse Niesen, Zaslav, Rjwilmsi, SmackBot, Harrigan, David Eppstein, Koko90, ImageRemovalBot, Justin W Smith, Tangi-tamma, JRN08, Yobot, Twri, LilHelpa, SlumdogAramis, Trappist the monk, Vengence313 and Anonymous: 10 • List of graphs Source: https://en.wikipedia.org/wiki/List_of_graphs?oldid=702638793 Contributors: Nonenmac, Michael Hardy, Booy- abazooka, Rich Farmbrough, BD2412, Rjwilmsi, Quuxplusone, Rmosler2100, Lyonsam, Misof, A3RO, David Eppstein, Koko90, Map- room, Austinmohr, Tangi-tamma, Cooldev.iitkgp, Twri, Thore Husfeldt, Lotje, Josve05a, ClueBot NG and Anonymous: 11 • Lévy family of graphs Source: https://en.wikipedia.org/wiki/L%C3%A9vy_family_of_graphs?oldid=702638914 Contributors: Zun- dark, Michael Hardy, Robinh, Andreas Kaufmann, BD2412, SmackBot, Myasuda, David Eppstein, SchreiberBike, McWomble, Yobot, Twri, ChrisGualtieri and Anonymous: 1 384 CHAPTER 104. ZERO-SYMMETRIC GRAPH

• Medial graph Source: https://en.wikipedia.org/wiki/Medial_graph?oldid=684373448 Contributors: Tomruen, Klemen Kocjancic, David Eppstein, Bender2k14, AnomieBOT, Schwede66, El Roih, Creeper jack1, Jeff Erickson and Anonymous: 1 • Median graph Source: https://en.wikipedia.org/wiki/Median_graph?oldid=707376763 Contributors: Giftlite, Igorpak, BD2412, Rjwilmsi, Headbomb, CharlotteWebb, Magioladitis, David Eppstein, Gambette, Fratrep, FghIJklm, Justin W Smith, Brewcrewer, Philippe Giab- banelli, Libcub, Addbot, SpellingBot, DisillusionedBitterAndKnackered, Kilom691, Citation bot, Howard McCay, LucienBOT, Citation bot 1, Vcunat, ClueBot NG, Helpful Pixie Bot, Jochen Burghardt and Anonymous: 4 • Moore graph Source: https://en.wikipedia.org/wiki/Moore_graph?oldid=710816666 Contributors: Booyabazooka, McKay, Giftlite, An- dreas Kaufmann, Zaslav, JHolman, Rjwilmsi, Maxal, Evilbu, Poulpy, Utopianheaven, Myasuda, Vanish2, Stdazi, David Eppstein, Robo- Tact, Eloz002, Mjaredd, Brookfield53045, Addbot, Citation bot, Twri, FrescoBot, John85, Citation bot 1, Rezabot, BG19bot, Gandalf- Graa, ChrisGualtieri, Tudor987 and Anonymous: 6 • Multipartite graph Source: https://en.wikipedia.org/wiki/Multipartite_graph?oldid=692324999 Contributors: Michael Hardy, Zaslav, Giraffedata, Malcolma, Myasuda, David Eppstein, MenoBot, ArticlesForCreationBot, Mogism, Jamesx12345, As acsi, Stannic, स्वागतम् and Anonymous: 1 • Null graph Source: https://en.wikipedia.org/wiki/Null_graph?oldid=702517606 Contributors: XJaM, Nonenmac, Booyabazooka, Jitse Niesen, Jni, MathMartin, Giftlite, Andreas Kaufmann, Paul August, Zaslav, Tompw, Oleg Alexandrov, Maxal, Derek Andrews, Incnis Mrsi, Melchoir, Movementarian, Nbarth, Jon Awbrey, JAnDbot, David Eppstein, JoergenB, Koko90, PaulTanenbaum, Jamelan, DumZi- BoT, Addbot, Webfarer, Yobot, Rubinbot, Twri, EmausBot, Ebrambot, CatParr, Helpful Pixie Bot, TricksterWolf, AdventurousSquirrel, Nathann.cohen, Bryanrutherford0 and Anonymous: 7 • Outerplanar graph Source: https://en.wikipedia.org/wiki/Outerplanar_graph?oldid=710979274 Contributors: AxelBoldt, Chowbok, Zaslav, Rjwilmsi, Headbomb, David Eppstein, Pichpich, MystBot, Addbot, Diwas, Helpful Pixie Bot and Anonymous: 4 • Overfull graph Source: https://en.wikipedia.org/wiki/Overfull_graph?oldid=702593021 Contributors: Michael Hardy, Andreas Kauf- mann, BD2412, RayAYang, David Eppstein, R'n'B, Jwuthe2, Yobot, Miym, FrescoBot and Anonymous: 3 • Panconnectivity Source: https://en.wikipedia.org/wiki/Panconnectivity?oldid=574580060 Contributors: Bearcat, Rich Farmbrough, R.e.b., David Eppstein, Tikiwont, Porchcorpter, Sławomir Biały, Bamyers99, M.aznaveh and Mark viking • Pancyclic graph Source: https://en.wikipedia.org/wiki/Pancyclic_graph?oldid=642975669 Contributors: Rjwilmsi, David Eppstein, El Roih, Mark viking and Anonymous: 1 • Partial cube Source: https://en.wikipedia.org/wiki/Partial_cube?oldid=702668573 Contributors: Bkell, BD2412, Rjwilmsi, Headbomb, David Eppstein, R'n'B, RjwilmsiBot and Anonymous: 1 • Partial k-tree Source: https://en.wikipedia.org/wiki/Partial_k-tree?oldid=539310304 Contributors: David Eppstein • Petersen family Source: https://en.wikipedia.org/wiki/Petersen_family?oldid=702552664 Contributors: Headbomb, David Eppstein, MystBot, Addbot, Nathann.cohen, Mark viking and Anonymous: 2 • Planar graph Source: https://en.wikipedia.org/wiki/Planar_graph?oldid=709608783 Contributors: AxelBoldt, LC~enwiki, Mav, Taral, Edemaine, Nonenmac, Heron, Tomo, Bdesham, Michael Hardy, Booyabazooka, Dominus, Shellreef, Rp, Gabbe, Nine Tail Fox, Eric119, Ahoerstemeier, Charles Matthews, Timwi, Dcoetzee, Dysprosia, Doradus, Taxman, Zero0000, McKay, Jaredwf, Altenmann, Romanm, MathMartin, DHN, Tobias Bergemann, Giftlite, Dbenbenn, Paul Pogonyshev, Everyking, Mellum, Jason Quinn, Alberto da Calvairate~enwiki, HorsePunchKid, Joyous!, McCart42, Trevor MacInnis, Corti, Svdb, Smyth, Paul August, Bender235, Zaslav, Brian0918, Bobo192, Jfraser, Thewayforward, ABCD, Burn, Oleg Alexandrov, Joriki, Linas, Magister Mathematicae, FreplySpang, Dpv, Jorunn, Rjwilmsi, NatusRoma, Dougluce, Marozols, FlaBot, Chris Pressey, [email protected], Mathbot, Hadaso, Chobot, YurikBot, Wavelength, Boticario, Dtrebbien, Bota47, LeonardoRob0t, PhS, Nsevs, SmackBot, Bobet, Zanetu, Taxipom, Jillbones, Tamfang, ManRabe~enwiki, Disavian, Lyonsam, Jamie King, Dilip rajeev, Lenoxus, Graph Theory page blanker, Gogo Dodo, Biruitorul, Headbomb, Libertyernie2, Urdutext, Cameron.walsh, Ubershmekel, A3nm, David Eppstein, Robin S, Afil, R'n'B, AstroHurricane001, Arunachalam.t, Koko90, LordAnubisBOT, Ged.R, Yecril, Kv75, Don4of4, Hagman, Cooperh, Debamf, OKBot, Anchor Link Bot, ImageRemovalBot, Wantnot, DFRussia, Snigbrook, Justin W Smith, Alexbot, M4gnum0n, Watchduck, Bender2k14, Sun Creator, AlexCornejo, Life of Riley, Myst- ,Yobot, Erel Segal, Citation bot, Hat600, Omnipaedista, Guillaume Damiand ,ماني ,Bot, Addbot, Some jerk on the Internet, SamatBot D'ohBot, Citation bot 1, Cutelyaware, EmausBot, Mastergreg82, JohnBoyerPhd, El Roih, Joel B. Lewis, Dainiak, Vinayak Pathak, JYBot, PassedTime, Rashteh, I Will Be Strong! and Anonymous: 94 • Platonic graph Source: https://en.wikipedia.org/wiki/Platonic_graph?oldid=702615480 Contributors: Tomruen, BD2412, Dr Greg, David Eppstein and Anonymous: 2 • Polytope graph Source: https://en.wikipedia.org/wiki/Convex_polytope?oldid=709607342 Contributors: Michael Hardy, Giftlite, David- Cary, MSGJ, Fropuff, Dratman, Waltpohl, Tomruen, Bender235, EmilJ, Marc van Woerkom, BD2412, Rjwilmsi, Wavelength, Mgnbar, Tetracube, Nbarth, E-Kartoffel, Lavaka, Steelpillow, David Eppstein, M-le-mot-dit, JohnBlackburne, Justin W Smith, Addbot, Yobot, Erel Segal, Twri, SteveWoolf, Kiefer.Wolfowitz, ZéroBot, Ben Shamos, MarcoMoellerHamburg, Lifeonahilltop, Helpful Pixie Bot, An- dreschulz, Panguipulli, Dexbot, Kaseypeesho and Anonymous: 8 • Prism graph Source: https://en.wikipedia.org/wiki/Prism_graph?oldid=702615296 Contributors: Michael Hardy, Tomruen, BD2412, Dr Greg and David Eppstein • Pseudoforest Source: https://en.wikipedia.org/wiki/Pseudoforest?oldid=708850243 Contributors: Zundark, Altenmann, Giftlite, An- dreas Kaufmann, Zaslav, Robert K S, Koavf, BorgQueen, Dan Hoey, Mhym, Headbomb, David Eppstein, Stephenchou0722, R'n'B, Koko90, Tcamps42, Webonfim, Jarble, Kilom691, Citation bot, Twri, Miym, Ltwp, Citation bot 1, Louistheran, ClueBot NG, Bibcode Bot, Bilorv, Inigolv and Anonymous: 5 • Quartic graph Source: https://en.wikipedia.org/wiki/Quartic_graph?oldid=702639273 Contributors: Tomo, Rich Farmbrough, Oleg Alexandrov, BD2412, Rjwilmsi, BiH, Nr9, David Eppstein, Watchduck, Twri and Erik9bot • Quasi-bipartite graph Source: https://en.wikipedia.org/wiki/Quasi-bipartite_graph?oldid=675445229 Contributors: Uncle G, Rjwilmsi, David Eppstein, Scift~enwiki, Shoessss, Daveagp, Yobot, Twri, Miym, Citation bot 1, Solomon7968 and Anonymous: 1 • Ramanujan graph Source: https://en.wikipedia.org/wiki/Ramanujan_graph?oldid=702639226 Contributors: The Anome, Michael Hardy, Charles Matthews, Cambyses, Qutezuce, BD2412, Bgwhite, Mhym, Kevin Milans, Cydebot, Rab V, Vanish2, David Eppstein, Robin S, DavidCBryant, TXiKiBoT, Addbot, DOI bot, Yobot, AnomieBOT, Twri, Citation bot 1, RobinK, Eransoko, BG19bot, Monkbot, After- before and Anonymous: 12 104.5. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 385

• Reeb graph Source: https://en.wikipedia.org/wiki/Reeb_graph?oldid=703443201 Contributors: Giftlite, BD2412, SmackBot, Derek farn, David Eppstein, Arcfrk, Addbot, Luckas-bot, Yobot, Ciphers, Twri, RjwilmsiBot, BG19bot, Teriyaki2109 and Anonymous: 4 • Regular graph Source: https://en.wikipedia.org/wiki/Regular_graph?oldid=706822291 Contributors: Andre Engels, XJaM, Tomo, Patrick, Michael Hardy, Mxn, McKay, Jaredwf, Altenmann, Gandalf61, MathMartin, Saforrest, Giftlite, Dbenbenn, Gene Ward Smith, Andreas Kaufmann, Arthena, Burn, Oleg Alexandrov, Oliphaunt, BD2412, Maxal, YurikBot, Dtrebbien, Bota47, Evilbu, Incnis Mrsi, AASoft, JoeKearney, Spiritia, Nikolas Karalis, Stdazi, David Eppstein, Yecril, TXiKiBoT, Rei-bot, Jamelan, Radagast3, Tim32, Hidro, DumZi- BoT, Tangi-tamma, Addbot, Luckas-bot, Yobot, Twri, Xqbot, GrouchoBot, Omnipaedista, FrescoBot, G.perarnau, Citation bot 1, Red- Bot, TobeBot, TheStrayCat, EmausBot, Sinuhe20, El Roih, Helpful Pixie Bot, Zouba, HandThumb and Anonymous: 21 • Scale-free network Source: https://en.wikipedia.org/wiki/Scale-free_network?oldid=704289337 Contributors: DavidLevinson, Heron, Dwheeler, Edward, Michael Hardy, Kku, Meekohi, CesarB, Yaronf, Cryoboy, Charles Matthews, Nickg, Peak, DavidCary, Harp, Wise- Woman, Lethe, Anville, Dratman, Kmote, SanderSpek~enwiki, ChaTo, TedPavlic, JimR, Longhair, Htmlism, 3mta3, Mdd, Cheezycrust, PAR, Andreala, Caesura, Dirac1933, Oleg Alexandrov, Dandv, Marudubshinki, Jshadias, MarkHudson, Rjwilmsi, Gareth McCaughan, JFromm, Mathbot, Gcalda, Madcoverboy, Conradl, Zwobot, FlyingPenguins, Gadget850, Fmccown, SmackBot, RDBrown, Gragus, Derlikous, Dreftymac, Danlev, Dlohcierekim, CmdrObot, Agathman, Gritzko, Myasuda, Ventania~enwiki, Cosmi, Headbomb, Anti- VandalBot, Joe Schmedley, Dougher, JAnDbot, Douglas R. White, Typochimp, David Eppstein, Onlynone, Gmagkots, Atddta, Dar- winPeacock, Pphaneuf, BarroColorado, Sandman2007, Econterms, PaulTanenbaum, Hannes Röst, Dirkbb, SieBot, Kl4m, DFRussia, Infoeco, Alexbot, SilvonenBot, Addbot, DOI bot, TutterMouse, Download, SamatBot, Echinoidea, Luckas-bot, Yobot, Nog33, Cita- tion bot, Miym, JonDePlume, Dkasthurirathna, Metasoarous, BenzolBot, RjwilmsiBot, Brteag00, WikitanvirBot, Massimo.franceschet, Smiling1126, ZéroBot, Bobsponj, Octochimps, Mouse20080706, Ugronugron, Helpful Pixie Bot, T214OU, Bibcode Bot, BG19bot, Tsn- diffopera, CitationCleanerBot, BattyBot, Alexandersaschawolff, ChrisGualtieri, Ophthalmol, Dexbot, Jaspermogg, 14GTR, FrigidNinja, Stamptrader, Monkbot, Malexmave, Hou710, Nurbudapest, Mil686, Slaplagne and Anonymous: 113 • Self-complementary graph Source: https://en.wikipedia.org/wiki/Self-complementary_graph?oldid=702638750 Contributors: Alten- mann, Giftlite, BD2412, Headbomb, Thadius856, David Eppstein, Kope, NuclearWarfare, Addbot, DSisyphBot, Mariana de El Mon- dongo, YFdyh-bot and Anonymous: 2 • Semi-symmetric graph Source: https://en.wikipedia.org/wiki/Semi-symmetric_graph?oldid=676563016 Contributors: Tomo, Michael Hardy, Silverfish, Ryan Reich, Mathbot, David Eppstein, Koko90, Maproom, Radagast3, Addbot, Citation bot, Twri, Miym, LucienBOT, Citation bot 1, EmausBot, ZéroBot and Anonymous: 3 • Series-parallel graph Source: https://en.wikipedia.org/wiki/Series-parallel_graph?oldid=706053792 Contributors: Edemaine, Edward, Altenmann, Bkell, Giftlite, Oliphaunt, Mamling, Ott2, Illythr, BetacommandBot, Hermel, D haggerty, David Eppstein, FJPB, Anna Lincoln, Aitias, Addbot, DOI bot, Yobot, Amirobot, Twri, Citation bot 1, Cheater no1, Deltahedron and Anonymous: 5 • Simplex graph Source: https://en.wikipedia.org/wiki/Simplex_graph?oldid=657375934 Contributors: Crasshopper, Headbomb, Huttarl, David Eppstein and Citation bot 1 • Skew-symmetric graph Source: https://en.wikipedia.org/wiki/Skew-symmetric_graph?oldid=609519537 Contributors: Giftlite, Zaslav, Oliphaunt, Ketiltrout, Rjwilmsi, Chris the speller, TheTito, David Eppstein, Addbot, Citation bot, Twri, Xqbot, Miym, Citation bot 1 and Helpful Pixie Bot • Small-world network Source: https://en.wikipedia.org/wiki/Small-world_network?oldid=704213110 Contributors: William Avery, Heron, Edward, Michael Hardy, Kku, Meekohi, Karada, Ronz, AugPi, Jitse Niesen, Bevo, Markus Krötzsch, Ds13, Dratman, Mboverload, Discospinster, Lejean2000, Paul August, Fenice, Davidgothberg, Tgr, Cheyinka, Linas, Mihai Damian, Kzollman, Stochata, XaosBits, BD2412, Rjwilmsi, NatusRoma, Mbutts, JFromm, Gurch, Debivort, Dbagnall, Wavelength, H005, JabberWok, Madcoverboy, Slar- son, Daniel Mietchen, Nethgirb, Cconnett, That Guy, From That Show!, SmackBot, Golbeck, Commander Keane bot, Drknexus, Chris the speller, Oli Filth, Uthbrian, Rludlow, Meson537, Ligulembot, Dankonikolic, Loodog, JHunterJ, Dragon guy, Myasuda, Gregbard, Headbomb, KConWiki, David Eppstein, User A1, PaulBHartzog, G.A.S, Urrameu, Eloz002, AKA MBG, DarwinPeacock, Fheyligh, KylieTastic, Pleasantville, Dggreen, Mkcmkc, Econterms, Wiae, Joetroll, Admiral Norton, Alexbot, Dwiddows, Terrillfrantz, Addbot, DOI bot, Obobskivich, EconoPhysicist, Netzwerkerin, SamatBot, Lightbot, AnomieBOT, Flopsy Mopsy and Cottonmouth, Citation bot, Xqbot, TRNiekras, Anna Frodesiak, Shadowjams, Urgos, Rafaelgoogle, ΙωάννηςΚαραμήτρος, Citation bot 1, Trappist the monk, Wolongzhiyong, Geostudent, Dangling Reference, Dennishouston, Helpful Pixie Bot, Bibcode Bot, BG19bot, Panchobook, Bereziny, CitationCleanerBot, Zujua, Nevik.R, BattyBot, WH98, ChrisGualtieri, DamascusGirlGeek, MidnightRequestLine, YiFeiBot, Schulllz, Lagoset, Monkbot, ClassicOnAStick, BxtrsChin, Shmall;D and Anonymous: 61 • Snark (graph theory) Source: https://en.wikipedia.org/wiki/Snark_(graph_theory)?oldid=693965420 Contributors: Taw, Heron, Cyde, Mxn, Charles Matthews, McKay, Kuszi, Bkell, Robinh, Dbenbenn, Gatta, Igorpak, Woohookitty, GregorB, Adking80, Rjwilmsi, Bhadani, Hairy Dude, Dmharvey, RussBot, Hakeem.gadi, Reyk, Evilbu, Scoutersig, SmackBot, Melchoir, Bigmike1020, Nbarth, Tsca.bot, Akri- asas, Jamie King, Ntsimp, After Midnight, Hermel, David Eppstein, Koko90, Maproom, Xvedejas, Gogobera, LokiClock, DOI bot, AnomieBOT, Citation bot, Twri, Omnipaedista, MathsPoetry, Nicko211, FrescoBot, Citation bot 1, Staszek Lem, Mikhail Ryazanov, Gabriel.c.drummond.cole, Helpful Pixie Bot, CitationCleanerBot, Mark viking, PC-XT and Anonymous: 18 • Split graph Source: https://en.wikipedia.org/wiki/Split_graph?oldid=711631516 Contributors: Michael Hardy, Altenmann, Zaslav, LOL, Adking80, Dtrebbien, Lyonsam, CmdrObot, Ntsimp, After Midnight, David Eppstein, Svick, Addbot, DOI bot, Tassedethe, Kilom691, Citation bot, Twri, Miym, Citation bot 1, Jonesey95, EmausBot, Darafsh, Monkbot, AnWiPaFr and Anonymous: 5 • Squaregraph Source: https://en.wikipedia.org/wiki/Squaregraph?oldid=565672076 Contributors: Tamfang, David Eppstein, Alexbot, Addbot, Citation bot 1, Foobarnix and Mark viking • Strangulated graph Source: https://en.wikipedia.org/wiki/Strangulated_graph?oldid=657227187 Contributors: David Eppstein • Strongly chordal graph Source: https://en.wikipedia.org/wiki/Strongly_chordal_graph?oldid=708561313 Contributors: Michael Hardy, Rjwilmsi, David Eppstein, Justin W Smith, Yobot, Bartosz Walczak, H.shahbazi, Frietjes, AnWiPaFr and Anonymous: 3 • Strongly regular graph Source: https://en.wikipedia.org/wiki/Strongly_regular_graph?oldid=706184056 Contributors: Michael Hardy, McKay, Giftlite, Dbenbenn, Tomruen, Qutezuce, Zaslav, Themusicgod1, Keenan Pepper, Brhaspati, Marozols, Evilbu, SmackBot, Blue- bot, J. Finkelstein, Gabn1, Citrus538, Nikolas Karalis, David Eppstein, Rocchini, Maproom, Radagast3, DFRussia, SchreiberBike, Myst- ,Luckas-bot, Yobot, AnomieBOT, Twri, RobinK, Vinay Madhusudanan, Wcherowi, Bazuz ,.עוזי ו ,Bot, Leen Droogendijk, Addbot BG19bot, ChrisGualtieri, Nathann.cohen, Teddyktchan and Anonymous: 21 • Subhamiltonian graph Source: https://en.wikipedia.org/wiki/Subhamiltonian_graph?oldid=683711259 Contributors: David Eppstein 386 CHAPTER 104. ZERO-SYMMETRIC GRAPH

• Symmetric graph Source: https://en.wikipedia.org/wiki/Symmetric_graph?oldid=702637319 Contributors: Tomo, Michael Hardy, Sil- verfish, Altenmann, Giftlite, YUL89YYZ, Zaslav, BD2412, Rjwilmsi, Myasuda, David Eppstein, Koko90, Maproom, VolkovBot, Rada- gast3, JackSchmidt, Alexbot, Addbot, Numbo3-bot, Luckas-bot, Twri, ArthurBot, Miym, D'ohBot, Citation bot 1, Arctanb, RobinK, ZéroBot, Helpful Pixie Bot and Anonymous: 2 • Table of simple cubic graphs Source: https://en.wikipedia.org/wiki/Table_of_simple_cubic_graphs?oldid=711071507 Contributors: Tomo, Giftlite, Rich Farmbrough, RDBury, Khazar, Headbomb, Stdazi, David Eppstein, FrescoBot, RjwilmsiBot, R. J. Mathar, Especially Lime and Anonymous: 7 • Threshold graph Source: https://en.wikipedia.org/wiki/Threshold_graph?oldid=705707719 Contributors: Zaslav, Oleg Alexandrov, Michael Slone, Lambiam, David Eppstein, VolkovBot, Mlampis, Addbot, Luckas-bot, Twri, DrilBot, Jonesey95 and Anonymous: 4 • Toroidal graph Source: https://en.wikipedia.org/wiki/Toroidal_graph?oldid=702671781 Contributors: Tomo, Michael Hardy, Meekohi, Silverfish, Rich Farmbrough, Gene Nygaard, Igorpak, BD2412, NatusRoma, OrphanBot, Mhym, Headbomb, David Eppstein, Rocchini, Koko90, Radagast3, Nicinic, MystBot, Addbot, Unzerlegbarkeit, Twri, Citation bot 1, Helpful Pixie Bot and Anonymous: 3 • Trellis (graph) Source: https://en.wikipedia.org/wiki/Trellis_(graph)?oldid=702636656 Contributors: HorsePunchKid, Andreas Kauf- mann, Bender235, BD2412, Bruyninc~enwiki, SmackBot, Mmernex, Pomte, Tikiwont, Don Quixote de la Mancha, Lambtron, Aquila78, Isheden, Paradrop and Anonymous: 1 • Triangle-free graph Source: https://en.wikipedia.org/wiki/Triangle-free_graph?oldid=711167079 Contributors: Booyabazooka, Rich Farmbrough, Igorpak, Rjwilmsi, RobertBorgersen, George100, Citrus538, Headbomb, David Eppstein, Koko90, DOI bot, Lightbot, Yobot, Fraggle81, AnomieBOT, Citation bot, Miym, Citation bot 1, RobinK, Xnn, RjwilmsiBot, ClueBot NG, Gruberan, Mathmasterx, Hnridder and Anonymous: 9 • Trivially perfect graph Source: https://en.wikipedia.org/wiki/Trivially_perfect_graph?oldid=706053302 Contributors: Lyonsam, David Eppstein, Citation bot, Citation bot 1, Pjpixi and Anonymous: 3 • Universal graph Source: https://en.wikipedia.org/wiki/Universal_graph?oldid=702636561 Contributors: Dominus, Silverfish, Aleph4, Oleg Alexandrov, BD2412, Davepape, Headbomb, KConWiki, David Eppstein, R'n'B, Addbot, DOI bot, Luckas-bot, Yobot, Twri, Citation bot 1, Jonesey95, El Roih and Helpful Pixie Bot • Vertex-transitive graph Source: https://en.wikipedia.org/wiki/Vertex-transitive_graph?oldid=702634233 Contributors: Tomo, Silver- fish, Charles Matthews, McKay, Phil Boswell, Jaredwf, Giftlite, Tomruen, BD2412, Rjwilmsi, Brighterorange, Mathbot, Algebraist, YurikBot, Evilbu, FRR~enwiki, Maksim-e~enwiki, Csab, Bluebot, Mhym, Ntsimp, Headbomb, David Eppstein, Koko90, Robert Illes, Radagast3, Addbot, Luckas-bot, Citation bot, Twri, Citation bot 1, RobinK, Xnn, ZéroBot, Solomon7968, PabloSpiga and Anonymous: 4 • Well-covered graph Source: https://en.wikipedia.org/wiki/Well-covered_graph?oldid=694129751 Contributors: Edward, Rjwilmsi, Chris the speller, David Eppstein, Justin W Smith and Frietjes • Zero-symmetric graph Source: https://en.wikipedia.org/wiki/Zero-symmetric_graph?oldid=642051529 Contributors: Tomruen, David Eppstein and (don't talk secrets)

104.5.2 Images

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• File:Apex_rhombic_dodecahedron.svg Source: https://upload.wikimedia.org/wikipedia/commons/7/7e/Apex_rhombic_dodecahedron. svg License: Public domain Contributors: Own work Original artist: David Eppstein • File:Apollonian-network.svg Source: https://upload.wikimedia.org/wikipedia/commons/8/80/Apollonian-network.svg License: CC0 Contributors: Own work Original artist: David Eppstein • File:Apollonian_gasket.svg Source: https://upload.wikimedia.org/wikipedia/commons/e/e6/Apollonian_gasket.svg License: GFDL Con- tributors: Own work Original artist: Time3000 • File:Asym-graph.PNG Source: https://upload.wikimedia.org/wikipedia/commons/0/08/Asym-graph.PNG License: CC BY-SA 3.0 Contributors: Own work Original artist: Mikkalai • File:Biclique_K_3_3.svg Source: https://upload.wikimedia.org/wikipedia/commons/f/f3/Biclique_K_3_3.svg License: CC BY-SA 3.0 Contributors: Own work Original artist: Koko90 • File:Biclique_K_3_5.svg Source: https://upload.wikimedia.org/wikipedia/commons/d/d6/Biclique_K_3_5.svg License: CC BY-SA 3.0 Contributors: Own work Original artist: Koko90 • File:Biclique_K_5_5.svg Source: https://upload.wikimedia.org/wikipedia/commons/5/5a/Biclique_K_5_5.svg License: CC BY-SA 3.0 Contributors: Own work Original artist: Koko90 • File:BiggsSmith.svg Source: https://upload.wikimedia.org/wikipedia/commons/3/32/BiggsSmith.svg License: Public domain Contribu- tors: Transferred from en.wikipedia to Commons. 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Original artist: en:User:Cburnett • File:Chessboard480.svg Source: https://upload.wikimedia.org/wikipedia/commons/d/d7/Chessboard480.svg License: CC0 Contribu- החבלן :tors: Own work Original artist • File:Chordal-graph.svg Source: https://upload.wikimedia.org/wikipedia/commons/3/34/Chordal-graph.svg License: Public domain Contributors: No machine-readable source provided. Own work assumed (based on copyright claims). Original artist: No machine- readable author provided. 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Original artist: ? • File:Complete_graph_K5.svg Source: https://upload.wikimedia.org/wikipedia/commons/c/cf/Complete_graph_K5.svg License: Pub- lic domain Contributors: Own work Original artist: David Benbennick wrote this ﬁle. • File:Complex_network_degree_distribution_of_random_and_scale-free.png Source: https://upload.wikimedia.org/wikipedia/commons/ 3/39/Complex_network_degree_distribution_of_random_and_scale-free.png License: Public domain Contributors: Drawn by the author Original artist: user:Sazaedo (ja:user:) • File:Convolutional_code_trellis_diagram.svg Source: https://upload.wikimedia.org/wikipedia/commons/a/a0/Convolutional_code_ trellis_diagram.svg License: Public domain Contributors: Own work Original artist: Qef • File:Cotree_and_cograph.svg Source: https://upload.wikimedia.org/wikipedia/commons/8/80/Cotree_and_cograph.svg License: Pub- lic domain Contributors: Own work Original artist: David Eppstein • File:Critical_graph_sample.svg Source: https://upload.wikimedia.org/wikipedia/commons/a/a8/Critical_graph_sample.svg License: CC BY-SA 3.0 Contributors: Own work Original artist: Claudio Rocchini 388 CHAPTER 104. ZERO-SYMMETRIC GRAPH

• File:Cube_graph.png Source: https://upload.wikimedia.org/wikipedia/commons/e/e1/Cube_graph.png License: CC-BY-SA-3.0 Con- tributors: ? Original artist: ? • File:Cube_retraction.svg Source: https://upload.wikimedia.org/wikipedia/commons/c/c3/Cube_retraction.svg License: Public domain Contributors: Own work Original artist: David Eppstein • File:Cubical_graph.png Source: https://upload.wikimedia.org/wikipedia/commons/3/3a/Cubical_graph.png License: CC BY-SA 4.0 Contributors: Own work Original artist: Tomruen • File:DeBruijn-as-line-digraph.svg Source: https://upload.wikimedia.org/wikipedia/commons/9/9d/DeBruijn-as-line-digraph.svg Li- cense: Public domain Contributors: self-made. Originally uploaded as en:Image:DeBruijn-as-line-digraph.png. Original artist: David Eppstein • File:Descartes_snark.png Source: https://upload.wikimedia.org/wikipedia/commons/2/27/Descartes_snark.png License: CC BY-SA 3.0 Contributors: Own work Original artist: Gabriel C. Drummond-Cole • File:Diamond_graph.svg Source: https://upload.wikimedia.org/wikipedia/commons/e/eb/Diamond_graph.svg License: CC BY-SA 3.0 Contributors: Own work Original artist: Koko90 • File:Directed_medial_graph_example.svg Source: https://upload.wikimedia.org/wikipedia/commons/6/69/Directed_medial_graph_ example.svg License: CC0 Contributors: \documentclass{standalone} \usepackage{tikz} \usetikzlibrary{arrows,backgrounds,positioning} \begin{document} \def\nodeDist{2.5cm} \def\arrowType{triangle 60} \def\fillColor{black!20} \def\position{0.6} \tikzstyle{origVertex} = [draw, blue, fill, shape=circle] \tikzstyle{newVertex} = [draw, red , fill, shape=circle] \tikzstyle{invisibleVertex} = [shape=circle] \tikzstyle{origEdge} = [blue] \tikzstyle{newEdge} = [red, densely dashed] \tikzstyle{invisibleEdge} = [draw opacity=0] \begin{tikzpicture}[node distance=\nodeDist,>=\arrowType,semithick] \node[origVertex] (0) {}; \node[origVertex] (1) [right of=0] {}; \node[origVertex] (2) [below of=1] {}; \node[origVertex] (3) [left of=2] {}; \path (0) edge[origEdge] node[invisibleVertex] (m0) {} (1) (1) edge[origEdge] node[invisibleVertex] (m1) {} (2) (2) edge[origEdge] node[invisibleVertex] (m2) {} (3) (3) edge[origEdge] node[invisibleVertex] (m3) {} (0); \path (m0) edge[invisibleEdge] (m1) edge[invisibleEdge, out= 45, in= 45, looseness=3, overlay] node[invisibleVertex, pos=\position] (e0) {} (m1) (m1) edge[invisibleEdge] (m2) edge[invisibleEdge, out= −45, in= −45, looseness=3, overlay] node[invisibleVertex, pos=\position] (e1) {} (m2) (m2) edge[invisibleEdge] (m3) edge[invisibleEdge, out=−135, in=−135, looseness=3, overlay] node[invisibleVertex, pos=\position] (e2) {} (m3) (m3) edge[invisibleEdge] (m0) edge[invisibleEdge, out= 135, in= 135, looseness=3, overlay] node[invisibleVertex, pos=\position] (e3) {} (m0); \node[invisibleVertex, below=0cm of e0] {}; \node[invisibleVertex, left =0cm of e1] {}; \node[invisibleVertex, above=0cm of e2] {}; \node[invisibleVertex, right=0cm of e3] {}; \end{tikzpicture} \qquad \begin{tikzpicture}[node distance=\nodeDist,>=\arrowType,semithick] \node[origVertex] (0) {}; \node[origVertex] (1) [right of=0] {}; \node[origVertex] (2) [below of=1] {}; \node[origVertex] (3) [left of=2] {}; \path (0) edge[origEdge] node[newVertex] (m0) {} (1) (1) edge[origEdge] node[newVertex] (m1) {} (2) (2) edge[origEdge] node[newVertex] (m2) {} (3) (3) edge[origEdge] node[newVertex] (m3) {} (0); \path (m0) edge[->, newEdge] (m1) edge[<-, newEdge, out= 45, in= 45, looseness=3, overlay] node[invisibleVertex, pos=\position] (e0) {} (m1) (m1) edge[->, newEdge] (m2) edge[<-, newEdge, out= −45, in= −45, looseness=3, overlay] node[invisibleVertex, pos=\position] (e1) {} (m2) (m2) edge[->, newEdge] (m3) edge[<-, newEdge, out=−135, in=−135, looseness=3, overlay] node[invisibleVertex, pos=\position] (e2) {} (m3) (m3) edge[- >, newEdge] (m0) edge[<-, newEdge, out= 135, in= 135, looseness=3, overlay] node[invisibleVertex, pos=\position] (e3) {} (m0); \node[invisibleVertex, below=0cm of e0] {}; \node[invisibleVertex, left =0cm of e1] {}; \node[invisibleVertex, above=0cm of e2] {}; \node[invisibleVertex, right=0cm of e3] {}; \begin{scope}[on background layer, overlay] \fill[fill=\fillColor] (m0. 45) to [out= 45, in= 45, looseness=3] (m1. 45) to (m1.center) to (m0.center) to (m0. 45); \fill[fill=\fillColor] (m1. −45) to [out= −45, in= −45, looseness=3] (m2. −45) to (m2.center) to (m1.center) to (m1. −45); \fill[fill=\fillColor] (m2.−135) to [out=−135, in=−135, looseness=3] (m3.−135) to (m3.center) to (m2.center) to (m2.−135); \fill[fill=\fillColor] (m3. 135) to [out= 135, in= 135, looseness=3] (m0. 135) to (m0.center) to (m3.center) to (m3. 135); \end{scope} \end{tikzpicture} \qquad \begin{tikzpicture}[node distance=\nodeDist,>=\arrowType,semithick] \node[invisibleVertex] (0) {}; \node[invisibleVertex] (1) [right of=0] {}; \node[invisibleVertex] (2) [below of=1] {}; \node[invisibleVertex] (3) [left of=2] {}; \path (0) edge[invisibleEdge] node[draw opacity=100, newVertex] 104.5. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 389

(m0) {} (1) (1) edge[invisibleEdge] node[draw opacity=100, newVertex] (m1) {} (2) (2) edge[invisibleEdge] node[draw opacity=100, newVertex] (m2) {} (3) (3) edge[invisibleEdge] node[draw opacity=100, newVertex] (m3) {} (0); \path (m0) edge[->, newEdge] (m1) edge[<-, newEdge, out= 45, in= 45, looseness=3, overlay] node[invisibleVertex, pos=\position] (e0) {} (m1) (m1) edge[->, newEdge] (m2) edge[<-, newEdge, out= −45, in= −45, looseness=3, overlay] node[invisibleVertex, pos=\position] (e1) {} (m2) (m2) edge[->, newEdge] (m3) edge[<-, newEdge, out=−135, in=−135, looseness=3, overlay] node[invisibleVertex, pos=\position] (e2) {} (m3) (m3) edge[- >, newEdge] (m0) edge[<-, newEdge, out= 135, in= 135, looseness=3, overlay] node[invisibleVertex, pos=\position] (e3) {} (m0); \node[invisibleVertex, below=0cm of e0] {}; \node[invisibleVertex, left =0cm of e1] {}; \node[invisibleVertex, above=0cm of e2] {}; \node[invisibleVertex, right=0cm of e3] {}; \begin{scope}[on background layer, overlay] \fill[fill=\fillColor] (m0. 45) to [out= 45, in= 45, looseness=3] (m1. 45) to (m1.center) to (m0.center) to (m0. 45); \fill[fill=\fillColor] (m1. −45) to [out= −45, in= −45, looseness=3] (m2. −45) to (m2.center) to (m1.center) to (m1. −45); \fill[fill=\fillColor] (m2.−135) to [out=−135, in=−135, looseness=3] (m3.−135) to (m3.center) to (m2.center) to (m2.−135); \fill[fill=\fillColor] (m3. 135) to [out= 135, in= 135, looseness=3] (m0. 135) to (m0.center) to (m3.center) to (m3. 135); \end{scope} \end{tikzpicture} \end{document}

Original artist: Self • File:Distance-hereditary_construction.svg Source: https://upload.wikimedia.org/wikipedia/commons/5/5d/Distance-hereditary_construction. svg License: Public domain Contributors: Own work Original artist: David Eppstein • File:Distance-hereditary_graph.svg Source: https://upload.wikimedia.org/wikipedia/commons/d/d1/Distance-hereditary_graph.svg Li- cense: Public domain Contributors: Own work Original artist: David Eppstein • File:Distributive_lattice_example.svg Source: https://upload.wikimedia.org/wikipedia/commons/d/d6/Distributive_lattice_example. svg License: Public domain Contributors: Own work Original artist: David Eppstein • File:Dodecahedral_graph.neato.svg Source: https://upload.wikimedia.org/wikipedia/commons/0/05/Dodecahedral_graph.neato.svg Li- cense: CC BY-SA 3.0 Contributors: Own work Original artist: Koko90 • File:Dodecahedron_schlegel_diagram.png Source: https://upload.wikimedia.org/wikipedia/commons/a/ad/Dodecahedron_schlegel_ diagram.png License: CC BY-SA 3.0 Contributors: Transferred from en.wikipedia Original artist: Tom Ruen (talk) • File:Dominating-set-2.svg Source: https://upload.wikimedia.org/wikipedia/commons/5/51/Dominating-set-2.svg License: CC BY-SA 3.0 Contributors: Own work Original artist: Miym • File:Dominating-set-reduction.svg Source: https://upload.wikimedia.org/wikipedia/commons/2/2c/Dominating-set-reduction.svg Li- cense: CC BY-SA 3.0 Contributors: Own work Original artist: Miym • File:Dominating-set.svg Source: https://upload.wikimedia.org/wikipedia/commons/e/e1/Dominating-set.svg License: CC BY-SA 3.0 Contributors: Own work Original artist: Miym • File:Dual_graphs.svg Source: https://upload.wikimedia.org/wikipedia/commons/1/15/Dual_graphs.svg License: Public domain Con- tributors: Transferred from en.wikipedia to Commons. Original artist: Booyabazooka at English Wikipedia • File:Edit-clear.svg Source: https://upload.wikimedia.org/wikipedia/en/f/f2/Edit-clear.svg License: Public domain Contributors: The Tango! Desktop Project. Original artist: The people from the Tango! project. And according to the meta-data in the ﬁle, speciﬁcally: “Andreas Nilsson, and Jakub Steiner (although minimally).” • File:Encoder_Disc_(3-Bit).svg Source: https://upload.wikimedia.org/wikipedia/commons/9/9b/Encoder_Disc_%283-Bit%29.svg Li- cense: Public domain Contributors: Replacement for Image:Encoder_disc.png Original artist: jjbeard • File:Factor_critical.svg Source: https://upload.wikimedia.org/wikipedia/commons/a/a9/Factor_critical.svg License: Public domain Con- tributors: Own work Original artist: David Eppstein • File:Finite-3-regular-graph-4-vertices.png Source: https://upload.wikimedia.org/wikipedia/commons/1/1d/Finite-3-regular-graph-4-vertices. png License: Public domain Contributors: Own work Original artist: Vassia Atanassova - Spiritia • File:Flower_snarkv.svg Source: https://upload.wikimedia.org/wikipedia/commons/6/61/Flower_snarkv.svg License: Public domain Contributors: No machine-readable source provided. Own work assumed (based on copyright claims). Original artist: No machine- readable author provided. Hakeem.gadi assumed (based on copyright claims). • File:Folkman_graph.svg Source: https://upload.wikimedia.org/wikipedia/commons/3/30/Folkman_graph.svg License: Public domain Contributors: Own work Original artist: David Eppstein • File:Forbidden_interval_subgraph.svg Source: https://upload.wikimedia.org/wikipedia/commons/e/e2/Forbidden_interval_subgraph. svg License: Public domain Contributors: Own work Original artist: David Eppstein • File:Forbidden_line_subgraphs.svg Source: https://upload.wikimedia.org/wikipedia/commons/6/6c/Forbidden_line_subgraphs.svg Li- cense: Public domain Contributors: Originally from en.wikipedia; description page is/was here. Original artist: Originally uploaded as a png image by David Eppstein at en.wikipedia. Redrawn as svg by Braindrain0000 at en.wikipedia • File:Friendship_graphs.svg Source: https://upload.wikimedia.org/wikipedia/commons/b/b1/Friendship_graphs.svg License: CC BY- SA 3.0 Contributors: Own work Original artist: Koko90 • File:Frucht_graph.dot.svg Source: https://upload.wikimedia.org/wikipedia/commons/3/36/Frucht_graph.dot.svg License: CC BY-SA 3.0 Contributors: Own work Original artist: Koko90 • File:Functional_graph.svg Source: https://upload.wikimedia.org/wikipedia/commons/d/d7/Functional_graph.svg License: Public domain Contributors: Own work Original artist: David Eppstein 390 CHAPTER 104. ZERO-SYMMETRIC GRAPH

• File:Goldner-Harary_graph.svg Source: https://upload.wikimedia.org/wikipedia/commons/9/96/Goldner-Harary_graph.svg License: Public domain Contributors: Own work Original artist: David Eppstein • File:Graph-Cartesian-product.svg Source: https://upload.wikimedia.org/wikipedia/commons/7/77/Graph-Cartesian-product.svg Li- cense: Public domain Contributors: Transferred from en.wikipedia to Commons. Original artist: David Eppstein at English Wikipedia • File:GraphY10W111EE4260094.jpg Source: https://upload.wikimedia.org/wikipedia/commons/1/1c/GraphY10W111EE4260094.jpg License: CC BY-SA 3.0 Contributors: Own work Original artist: R. J. Mathar • File:GraphY10W79EE3472233.jpg Source: https://upload.wikimedia.org/wikipedia/commons/0/0e/GraphY10W79EE3472233.jpg Li- cense: CC BY-SA 3.0 Contributors: Own work Original artist: R. J. Mathar • File:GraphY10W81EE3497449.jpg Source: https://upload.wikimedia.org/wikipedia/commons/3/3d/GraphY10W81EE3497449.jpg Li- cense: CC BY-SA 3.0 Contributors: Own work Original artist: R. J. Mathar • File:GraphY10W81EE3677120.jpg Source: https://upload.wikimedia.org/wikipedia/commons/a/a5/GraphY10W81EE3677120.jpg Li- cense: CC BY-SA 3.0 Contributors: Own work Original artist: R. J. Mathar • File:GraphY10W82EE3600347.jpg Source: https://upload.wikimedia.org/wikipedia/commons/d/d7/GraphY10W82EE3600347.jpg Li- cense: CC BY-SA 3.0 Contributors: Own work Original artist: R. J. Mathar • File:GraphY10W83EE3701785.jpg Source: https://upload.wikimedia.org/wikipedia/commons/f/f0/GraphY10W83EE3701785.jpg Li- cense: CC BY-SA 3.0 Contributors: Own work Original artist: R. J. Mathar • File:GraphY10W84EE3801880.jpg Source: https://upload.wikimedia.org/wikipedia/commons/f/f9/GraphY10W84EE3801880.jpg Li- cense: CC BY-SA 3.0 Contributors: Own work Original artist: R. J. 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• File:Medial_graph_example.svg Source: https://upload.wikimedia.org/wikipedia/commons/6/69/Medial_graph_example.svg License: CC0 Contributors: \documentclass{article} \thispagestyle{empty} \usepackage{tikz} \def\nodeDist{2.5cm} \tikzstyle{origVertex} = [draw, blue, ﬁll, shape=circle] \tikzstyle{newVertex} = [draw, red, ﬁll, shape=circle] \tikzstyle{invisibleVertex} = [shape=circle] \tikzstyle{origEdge} = [blue] \tikzstyle{newEdge} = [red, densely dashed] \tikzstyle{invisibleEdge} = [draw opacity=0] \begin{document} \begin{tikzpicture}[node distance=\nodeDist,semithick] \node[origVertex] (0) {}; \node[origVertex] (1) [right of=0] {}; \node[origVertex] (2) [above of=0] {}; \node[origVertex] (3) [above of=1] {}; \node[origVertex] (4) [above of=2] {}; \path (0) edge[origEdge, out=−45, in=−135] node[invisibleVertex] (m0) {} (1) edge[origEdge, out= 45, in= 135] node[invisibleVertex] (m1) {} (1) edge[origEdge] node[invisibleVertex] (m2) {} (2) (1) edge[origEdge] node[invisibleVertex] (m3) {} (3) (2) edge[origEdge] node[invisibleVertex] (m4) {} (3) edge[origEdge] node[invisibleVertex] (m5) {} (4) (3) edge[origEdge, out=125, in= 55, looseness=30] node[invisibleVertex] (m6) {} (3); \path (m0) edge[invisibleEdge, out= 135, in=−135] (m1) edge[invisibleEdge, out= 45, in= −45] (m1) edge[invisibleEdge, out=−145, in=−135, looseness=1.7] (m2) edge[invisibleEdge, out= −35, in= −45, looseness=1.7] (m3) (m1) edge[invisibleEdge] (m2) edge[invisibleEdge] (m3) (m2) edge[invisibleEdge] (m4) edge[invisibleEdge, out= 135, in=−135] (m5) (m3) edge[invisibleEdge] (m4) edge[invisibleEdge, out= 45, in= 15] (m6) (m4) edge[invisibleEdge] (m5) edge[invisibleEdge, out= 90, in= 165] (m6) (m5) edge[invisibleEdge, out= 125, in= 55, looseness=30] (m5) (m6) edge[invisibleEdge, out=−125, in= −55, looseness=15] (m6); \end{tikzpicture} \begin{tikzpicture}[node distance=\nodeDist,semithick] \node[origVertex] (0) {}; \node[origVertex] (1) [right of=0] {}; \node[origVertex] (2) [above of=0] {}; \node[origVertex] (3) [above of=1] {}; \node[origVertex] (4) [above of=2] {}; \path (0) edge[origEdge, out=−45, in=−135] node[newVertex] (m0) {} (1) edge[origEdge, out= 45, in= 135] node[newVertex] (m1) {} (1) edge[origEdge] node[newVertex] (m2) {} (2) (1) edge[origEdge] node[newVertex] (m3) {} (3) (2) edge[origEdge] node[newVertex] (m4) {} (3) edge[origEdge] node[newVertex] (m5) {} (4) (3) edge[origEdge, out=125, in= 55, looseness=30] node[newVertex] (m6) {} (3); \path (m0) edge[newEdge, out= 135, in=−135] (m1) edge[newEdge, out= 45, in= −45] (m1) edge[newEdge, out=−145, in=−135, looseness=1.7] (m2) edge[newEdge, out= −35, in= −45, looseness=1.7] (m3) (m1) edge[newEdge] (m2) edge[newEdge] (m3) (m2) edge[newEdge] (m4) edge[newEdge, out= 135, in=−135] (m5) (m3) edge[newEdge] (m4) edge[newEdge, out= 45, in= 15] (m6) (m4) edge[newEdge] (m5) edge[newEdge, out= 90, in= 165] (m6) (m5) edge[newEdge, out= 125, in= 55, looseness=30] (m5) (m6) edge[newEdge, out=−125, in= −55, looseness=15] (m6); \end{tikzpicture} \begin{tikzpicture}[node distance=\nodeDist,semithick] \node[invisibleVertex] (0) {}; \node[invisibleVertex] (1) [right of=0] {}; \node[invisibleVertex] (2) [above of=0] {}; \node[invisibleVertex] (3) [above of=1] {}; \node[invisibleVertex] (4) [above of=2] {}; \path (0) edge[invisibleEdge, out=−45, in=−135] node[newVertex] (m0) {} (1) edge[invisibleEdge, out= 45, in= 135] node[newVertex] (m1) {} (1) edge[invisibleEdge] node[newVertex] (m2) {} (2) (1) edge[invisibleEdge] node[newVertex] (m3) {} (3) (2) edge[invisibleEdge] node[newVertex] (m4) {} (3) edge[invisibleEdge] node[newVertex] (m5) {} (4) (3) edge[invisibleEdge, out=125, in= 55, looseness=30] node[newVertex] (m6) {} (3); \path (m0) edge[newEdge, out= 135, in=−135] (m1) edge[newEdge, out= 45, in= −45] (m1) edge[newEdge, out=−145, in=−135, looseness=1.7] (m2) edge[newEdge, out= −35, in= −45, looseness=1.7] (m3) (m1) edge[newEdge] (m2) edge[newEdge] (m3) (m2) edge[newEdge] (m4) edge[newEdge, out= 135, in=−135] (m5) (m3) edge[newEdge] (m4) edge[newEdge, out= 45, in= 15] (m6) (m4) edge[newEdge] (m5) edge[newEdge, out= 90, in= 165] (m6) (m5) edge[newEdge, out= 125, in= 55, looseness=30] (m5) (m6) edge[newEdge, out=−125, in= −55, looseness=15] (m6); \end{tikzpicture} \end{document}

Original artist: Self • File:Median_from_triangle-free.svg Source: https://upload.wikimedia.org/wikipedia/commons/e/e0/Median_from_triangle-free.svg License: Public domain Contributors: Own work Original artist: David Eppstein • File:Median_graph.svg Source: https://upload.wikimedia.org/wikipedia/commons/5/56/Median_graph.svg License: Public domain Con- tributors: Own work Original artist: David Eppstein • File:Moebius-ladder-16.svg Source: https://upload.wikimedia.org/wikipedia/commons/e/e3/Moebius-ladder-16.svg License: Public domain Contributors: self-made; originally uploaded as PNG to en:Image:Moebius-ladder-16.png Original artist: David Eppstein • File:Moser_spindle_pseudotriangulation.svg Source: https://upload.wikimedia.org/wikipedia/commons/a/af/Moser_spindle_pseudotriangulation. svg License: CC0 Contributors: Own work Original artist: David Eppstein • File:Nonisomorphic_medial_graph_example.svg Source: https://upload.wikimedia.org/wikipedia/commons/a/a7/Nonisomorphic_medial_ graph_example.svg License: CC0 Contributors: \documentclass{article} \thispagestyle{empty} \usepackage[margin=0in]

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\end{tikzpicture} \begin{tikzpicture}[node distance=\nodeDist,semithick] \node[origVertex] (0) {}; \node[invisibleVertex] (1) [below of=0] {}; \node[origVertex] (2) [left of=1] {}; \node[origVertex] (3) [right of=1] {}; \node[origVertex] (4) [below of=1] {}; \node[origVertex] (5) [below left of=4] {}; \node[origVertex] (6) [below right of=4] {}; \path (0) edge[origEdge] node[newVertex] (m0) {} (2) edge[origEdge] node[newVertex] (m1) {} (3) (2) edge[origEdge] node[newVertex] (m2) {} (4) (3) edge[origEdge] node[newVertex] (m3) {} 396 CHAPTER 104. ZERO-SYMMETRIC GRAPH

(4) (4) edge[origEdge] node[newVertex] (m4) {} (5) edge[origEdge] node[newVertex] (m5) {} (6); \path (m0) edge[newEdge, out= 90, in= 90, looseness=2] (m1) edge[newEdge] (m1) edge[newEdge, out= 180, in= 180, looseness=2] (m2) edge[newEdge] (m2) (m1) edge[newEdge, out= 0, in= 0, looseness=2] (m3) edge[newEdge] (m3) (m2) edge[newEdge] (m3) edge[newEdge] (m4) (m3) edge[newEdge] (m5) (m4) edge[newEdge, out=−170, in=−100, looseness=30] (m4) edge[newEdge] (m5) (m5) edge[newEdge, out= −80, in= −10, looseness=30] (m5); \end{tikzpicture} \begin{tikzpicture}[node distance=\nodeDist,semithick] \node[invisibleVertex] (0) {}; \node[invisibleVertex] (1) [below of=0] {}; \node[invisibleVertex] (2) [left of=1] {}; \node[invisibleVertex] (3) [right of=1] {}; \node[invisibleVertex] (4) [below of=1] {}; \node[invisibleVertex] (5) [below left of=4] {}; \node[invisibleVertex] (6) [below right of=4] {}; \path (0) edge[invisibleEdge] node[newVertex] (m0) {} (2) edge[invisibleEdge] node[newVertex] (m1) {} (3) (2) edge[invisibleEdge] node[newVertex] (m2) {} (4) (3) edge[invisibleEdge] node[newVertex] (m3) {} (4) (4) edge[invisibleEdge] node[newVertex] (m4) {} (5) edge[invisibleEdge] node[newVertex] (m5) {} (6); \path (m0) edge[newEdge, out= 90, in= 90, looseness=2] (m1) edge[newEdge] (m1) edge[newEdge, out= 180, in= 180, looseness=2] (m2) edge[newEdge] (m2) (m1) edge[newEdge, out= 0, in= 0, looseness=2] (m3) edge[newEdge] (m3) (m2) edge[newEdge] (m3) edge[newEdge] (m4) (m3) edge[newEdge] (m5) (m4) edge[newEdge, out=−170, in=−100, looseness=30] (m4) edge[newEdge] (m5) (m5) edge[newEdge, out= −80, in= −10, looseness=30] (m5); \end{tikzpicture} \end{document}

Original artist: Self • File:Nonplanar_no_subgraph_K_3_3.svg Source: https://upload.wikimedia.org/wikipedia/en/d/dc/Nonplanar_no_subgraph_K_3_3. svg License: CC-BY-SA-3.0 Contributors: Self-made Original artist: Daniel M. Short • File:OEISicon_light.svg Source: https://upload.wikimedia.org/wikipedia/commons/d/d8/OEISicon_light.svg License: Public domain Contributors: Own work Original artist: Watchduck (a.k.a. Tilman Piesk) • File:Octagonal_antiprismatic_graph.png Source: https://upload.wikimedia.org/wikipedia/commons/c/c5/Octagonal_antiprismatic_ graph.png License: CC BY-SA 4.0 Contributors: Own work Original artist: Tomruen • File:Octagonal_prismatic_graph.png Source: https://upload.wikimedia.org/wikipedia/commons/d/d8/Octagonal_prismatic_graph.png License: CC BY-SA 4.0 Contributors: Own work Original artist: Tomruen • File:Odd_Cycle_Transversal_of_size_2.png Source: https://upload.wikimedia.org/wikipedia/commons/d/d4/Odd_Cycle_Transversal_ of_size_2.png License: CC BY-SA 3.0 Contributors: Own work Original artist: PG Drange • File:Paley13.svg Source: https://upload.wikimedia.org/wikipedia/commons/0/05/Paley13.svg License: Public domain Contributors: Trans- ferred from en.wikipedia to Commons. Original artist: David Eppstein at English Wikipedia • File:Pancyclic_octahedron.svg Source: https://upload.wikimedia.org/wikipedia/commons/7/77/Pancyclic_octahedron.svg License: Pub- lic domain Contributors: Own work Original artist: David Eppstein • File:Partial_3-tree_forbidden_minors.svg Source: https://upload.wikimedia.org/wikipedia/commons/7/73/Partial_3-tree_forbidden_ minors.svg License: Public domain Contributors: Own work Original artist: David Eppstein • File:Pentagonal_antiprismatic_graph.png Source: https://upload.wikimedia.org/wikipedia/commons/b/b1/Pentagonal_antiprismatic_ graph.png License: CC BY-SA 4.0 Contributors: Own work Original artist: Tomruen • File:Pentagonal_prismatic_graph.png Source: https://upload.wikimedia.org/wikipedia/commons/a/aa/Pentagonal_prismatic_graph. png License: CC BY-SA 4.0 Contributors: Own work Original artist: Tomruen • File:Period-3-graph.svg Source: https://upload.wikimedia.org/wikipedia/commons/2/20/Period-3-graph.svg License: Public domain Contributors: Transferred from en.wikipedia to Commons. Original artist: David Eppstein at English Wikipedia • File:Petersen-as-Moore.svg Source: https://upload.wikimedia.org/wikipedia/commons/0/00/Petersen-as-Moore.svg License: Public domain Contributors: self-made; originally uploaded as PNG to en:Image:Petersen-as-Moore.png Original artist: David Eppstein • File:Petersen1_tiny.svg Source: https://upload.wikimedia.org/wikipedia/commons/9/91/Petersen1_tiny.svg License: CC BY-SA 3.0 Contributors: Own work by uploader based on http://en.wikipedia.org/wiki/File:Heawood_Graph.svg Original artist: Leshabirukov • File:Petersen_family.svg Source: https://upload.wikimedia.org/wikipedia/commons/0/04/Petersen_family.svg License: Public domain Contributors: Own work Original artist: David Eppstein • File:Polyhedral_schlegel_diagrams.svg Source: https://upload.wikimedia.org/wikipedia/commons/7/7e/Polyhedral_schlegel_diagrams. svg License: Public domain Contributors: en:Image:Polyhedral schlegel diagrams.png Original artist: Traced by User:Stannered • File:Poset_et_graphe_de_comparabilité.svg Source: https://upload.wikimedia.org/wikipedia/commons/2/29/Poset_et_graphe_de_comparabilit% C3%A9.svg License: CC BY-SA 3.0 Contributors: Own work Original artist: Kilom691 • File:Pseudoforest.svg Source: https://upload.wikimedia.org/wikipedia/commons/8/85/Pseudoforest.svg License: Public domain Con- tributors: Own work Original artist: David Eppstein • File:Pyramid_face_lattice.svg Source: https://upload.wikimedia.org/wikipedia/commons/4/4c/Pyramid_face_lattice.svg License: Pub- lic domain Contributors: Own work Original artist: David Eppstein • File:Question_book-new.svg Source: https://upload.wikimedia.org/wikipedia/en/9/99/Question_book-new.svg License: Cc-by-sa-3.0 Contributors: Created from scratch in Adobe Illustrator. Based on Image:Question book.png created by User:Equazcion Original artist: Tkgd2007 104.5. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 397

• File:Question_dropshade.png Source: https://upload.wikimedia.org/wikipedia/commons/d/dd/Question_dropshade.png License: Pub- lic domain Contributors: Image created by JRM Original artist: JRM • File:Random_graph_gephi.png Source: https://upload.wikimedia.org/wikipedia/commons/a/ac/Random_graph_gephi.png License: CC BY 3.0 Contributors: Own work Original artist: Schulllz • File:Repeated_diamond_graph.svg Source: https://upload.wikimedia.org/wikipedia/commons/7/75/Repeated_diamond_graph.svg Li- cense: Public domain Contributors: Self-made. Layout copied from a raster image of the same subgraph by Tangi-tamma. Original artist: David Eppstein • File:Scale-free_network_sample.png Source: https://upload.wikimedia.org/wikipedia/commons/7/77/Scale-free_network_sample.png License: CC-BY-SA-3.0 Contributors: ? Original artist: ? • File:Self-complementary_NZ_graph.svg Source: https://upload.wikimedia.org/wikipedia/commons/e/e8/Self-complementary_NZ_ graph.svg License: Public domain Contributors: Own work Original artist: David Eppstein • File:Series_parallel_composition.svg Source: https://upload.wikimedia.org/wikipedia/commons/d/da/Series_parallel_composition.svg License: Public domain Contributors: Transferred from en.wikipedia to Commons. Original artist: David Eppstein at English Wikipedia • File:Simple-bipartite-graph.svg Source: https://upload.wikimedia.org/wikipedia/commons/e/e8/Simple-bipartite-graph.svg License: Public domain Contributors: Own work Original artist: MistWiz • File:Simplex_graph.svg Source: https://upload.wikimedia.org/wikipedia/commons/4/4c/Simplex_graph.svg License: Public domain Contributors: Own work by uploader. Based on File:Median from triangle-free.svg. 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I copy pictures from http://www.graphclasses.org/smallgraphs.html#nodes3 Original artist: Webonﬁm • File:Thomassen_girth-3_hypohamiltonian.svg Source: https://upload.wikimedia.org/wikipedia/commons/e/e4/Thomassen_girth-3_ hypohamiltonian.svg License: Public domain Contributors: Own work Original artist: David Eppstein • File:Threshold_graph.png Source: https://upload.wikimedia.org/wikipedia/en/0/00/Threshold_graph.png License: PD Contributors: ? Original artist: ? • File:Toroidal_graph_sample.gif Source: https://upload.wikimedia.org/wikipedia/commons/9/97/Toroidal_graph_sample.gif License: CC BY 2.5 Contributors: Own work Original artist: Claudio Rocchini • File:Tree_decomposition.svg Source: https://upload.wikimedia.org/wikipedia/commons/a/a7/Tree_decomposition.svg License: Public domain Contributors: Own work Original artist: David Eppstein • File:Tree_median.svg Source: https://upload.wikimedia.org/wikipedia/commons/2/29/Tree_median.svg License: Public domain Con- tributors: Own work Original artist: David Eppstein • File:Triakistetrahedron.jpg Source: https://upload.wikimedia.org/wikipedia/commons/d/de/Triakistetrahedron.jpg License: CC-BY- SA-3.0 Contributors: ? Original artist: ? • File:Triangular_prism_as_halin_graph.svg Source: https://upload.wikimedia.org/wikipedia/commons/f/fb/Triangular_prism_as_halin_ graph.svg License: CC BY-SA 3.0 Contributors: Own work Original artist: MathsPoetry • File:Triangular_prismatic_graph.png Source: https://upload.wikimedia.org/wikipedia/commons/d/de/Triangular_prismatic_graph.png License: CC BY-SA 4.0 Contributors: Own work Original artist: Tomruen • File:Triangulation_3-coloring.svg Source: https://upload.wikimedia.org/wikipedia/commons/c/c2/Triangulation_3-coloring.svg License: Public domain Contributors: Own work Original artist: David Eppstein • File:Trivially_perfect_graph.svg Source: https://upload.wikimedia.org/wikipedia/commons/f/fa/Trivially_perfect_graph.svg License: Public domain Contributors: Own work Original artist: David Eppstein • File:Tuncated_tetrahedral_graph.png Source: https://upload.wikimedia.org/wikipedia/commons/c/c3/Tuncated_tetrahedral_graph. png License: CC BY-SA 4.0 Contributors: Own work Original artist: Tomruen • File:Turan_13-4.svg Source: https://upload.wikimedia.org/wikipedia/commons/1/1b/Turan_13-4.svg License: Public domain Contrib- utors: Transferred from en.wikipedia to Commons. Original artist: SVG image created by Braindrain0000 at en.wikipedia, based on previous public-domain PNG image en::Image:Turan-13-4.png by David Eppstein • File:Tutte_eight_cage.svg Source: https://upload.wikimedia.org/wikipedia/commons/4/41/Tutte_eight_cage.svg License: Public domain Contributors: ? Original artist: ? 398 CHAPTER 104. ZERO-SYMMETRIC GRAPH

• File:Well-covered_graph.svg Source: https://upload.wikimedia.org/wikipedia/commons/9/94/Well-covered_graph.svg License: CC0 Contributors: Own work Modiﬁed from one of the graphs in File:Petersen family.svg Original artist: David Eppstein • File:Wheel_graphs.svg Source: https://upload.wikimedia.org/wikipedia/commons/2/22/Wheel_graphs.svg License: Public domain Con- tributors: Self-made. Originally uploaded as PNG to en.wikipedia; description page is/was here. Original artist: David Eppstein • File:Wiki_letter_w.svg Source: https://upload.wikimedia.org/wikipedia/en/6/6c/Wiki_letter_w.svg License: Cc-by-sa-3.0 Contributors: ? Original artist: ? • File:Y10W75EE3421829.jpg Source: https://upload.wikimedia.org/wikipedia/commons/9/99/Y10W75EE3421829.jpg License: CC BY-SA 3.0 Contributors: Own work Original artist: R. J. Mathar • File:Y10W84EE3625442.jpg Source: https://upload.wikimedia.org/wikipedia/commons/2/2e/Y10W84EE3625442.jpg License: CC BY-SA 3.0 Contributors: Own work Original artist: R. J. Mathar • File:Y10W87EE3769671.jpg Source: https://upload.wikimedia.org/wikipedia/commons/f/f7/Y10W87EE3769671.jpg License: CC BY- SA 3.0 Contributors: Own work Original artist: R. J. Mathar • File:Y10W90EE3890980.jpg Source: https://upload.wikimedia.org/wikipedia/commons/d/da/Y10W90EE3890980.jpg License: CC BY-SA 3.0 Contributors: Own work Original artist: R. J. Mathar • File:Y12W129EE4170908.jpg Source: https://upload.wikimedia.org/wikipedia/commons/8/87/Y12W129EE4170908.jpg License: CC BY-SA 3.0 Contributors: Own work Original artist: R. J. Mathar • File:Y12W130EE4102128.jpg Source: https://upload.wikimedia.org/wikipedia/commons/6/6a/Y12W130EE4102128.jpg License: CC BY-SA 3.0 Contributors: Own work Original artist: R. J. Mathar • File:Y12W132EE4105212.jpg Source: https://upload.wikimedia.org/wikipedia/commons/c/c0/Y12W132EE4105212.jpg License: CC BY-SA 3.0 Contributors: Own work Original artist: R. J. Mathar • File:Y12W134EE4166461.jpg Source: https://upload.wikimedia.org/wikipedia/commons/e/ef/Y12W134EE4166461.jpg License: CC BY-SA 3.0 Contributors: Own work Original artist: R. J. Mathar • File:Y12W135EE4325057.jpg Source: https://upload.wikimedia.org/wikipedia/commons/5/52/Y12W135EE4325057.jpg License: CC BY-SA 3.0 Contributors: Own work Original artist: R. J. Mathar • File:Y12W136EE4145861.jpg Source: https://upload.wikimedia.org/wikipedia/commons/d/d8/Y12W136EE4145861.jpg License: CC BY-SA 3.0 Contributors: Own work Original artist: R. J. Mathar • File:Y12W138EE4225614.jpg Source: https://upload.wikimedia.org/wikipedia/commons/1/15/Y12W138EE4225614.jpg License: CC BY-SA 3.0 Contributors: Own work Original artist: R. J. Mathar • File:Y8W44EE3003607.jpg Source: https://upload.wikimedia.org/wikipedia/commons/9/98/Y8W44EE3003607.jpg License: CC BY- SA 3.0 Contributors: Own work Original artist: R. J. Mathar • File:Y8W46EE3097135.jpg Source: https://upload.wikimedia.org/wikipedia/commons/f/f8/Y8W46EE3097135.jpg License: CC BY- SA 3.0 Contributors: Own work Original artist: R. J. Mathar • File:Y8W50EE3373868.jpg Source: https://upload.wikimedia.org/wikipedia/commons/4/40/Y8W50EE3373868.jpg License: CC BY- SA 3.0 Contributors: Own work Original artist: R. J. Mathar

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