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

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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.
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