Appendix A

Graph Theory Background

A graph G = (V,E) consists of two finite sets V and E. The elements of V are called the vertices and the elements of E the edges of G. Each edge is a pair of vertices. Graphs have natural graphical representations in which each is rep- resented by a point and each edge by a line connecting two points. Figure A.1 represents the graph G = (V,E) with vertex set V = {1, 2, 3, 4, 5} and edge set E = {12, 23, 34, 45}. 1 2 3 4 5

Figure A.1: Graph G = (V,E) with V = {1, 2, 3, 4, 5} and E = {12, 23, 34, 45}

By altering the definition, we can obtain different types of graphs. For instance,

• by replacing the set E with a set of ordered pairs of vertices, we obtain a (digraph for short), also known as an oriented graph. • by allowing E to contain both directed and undirected edges, we obtain a mixed graph. • by allowing repeated elements in the set of edges, i.e., by replacing E with a multiset, we obtain a multigraph. • by allowing edges to connect a vertex to itself (a loop), we obtain a pseudo- graph. • by allowing the edges to be arbitrary subsets of vertices, not necessarily of size two, we obtain a hypergraph.

© Springer International Publishing Switzerland 2015 231 S. Kitaev, V. Lozin, Words and Graphs, Monographs in Theoretical Computer Science. An EATCS Series, DOI 10.1007/978-3-319-25859-1 232 Appendix A. Background

• by allowing V and E to be infinite sets, we obtain an infinite graph.

Definition A.0.1. A simple graph is a finite undirected graph without loops and multiple edges.

All graphs in this book are simple, unless stated otherwise.

A.1 Terminology and Notation

For a graph G, we denote by V (G) and E(G) the set of vertices and the set of edges of G, respectively. Definition A.1.1. Let u, v be two vertices of a graph G.

• If uv ∈ E(G), then u, v are said to be adjacent, in which case we also say that u is connected to v or u is a neighbour of v. If uv ̸∈ E(G), then u and v are nonadjacent (not connected, non-neighbours).

• The neighbourhood of a vertex v ∈ V (G), denoted N(v), is the set of vertices adjacent to v, i.e. N(v) = {u ∈ V (G) | vu ∈ E(G)}.

• If e = uv is an edge of G, then e is incident to u and v. We also say that u and v are the endpoints of e.

• The of v ∈ V (G), denoted deg(v), is the number of edges incident to v. Alternatively, deg(v) = |N(v)|. Definition A.1.2. The complement of a graph G = (V,E) is a graph with vertex set V and edge set E′ such that e ∈ E′ if and only if e ̸∈ E. The complement of a graph G is denoted G or co-G.

Definition A.1.3. Given two graphs G1 = (V1,E1) and G2 = (V2,E2), the graph G1 is said to be

• a subgraph of G2 if V1 ⊆ V2 and E1 ⊆ E2, i.e. G1 can be obtained from G2 by deleting some vertices (with all incident edges) and some edges;

• a spanning subgraph of G2 if V1 = V2, i.e. G1 can be obtained from G2 by deleting some edges but not vertices;

• an of G2 if every edge of G2 with both endpoints in V1 is also an edge of G1, i.e. G1 can be obtained from G2 by deleting some vertices (with all incident edges) but not edges. A.1 Terminology and Notation 233

Definition A.1.4. In a graph,

• a walk is a sequence of (not necessarily distinct) vertices v1, v2, . . . , vk such that vi is adjacent to vi+1 for each i = 1, . . . , k − 1. A walk can alternatively be described by the set of its edges v1v2, . . . , vk−1vk. The length of a walk is the number of its edges. A walk is closed if v1 = vk.

• a trail is a walk with no repeated edges.

• a path is a walk with no repeated vertices.

• a is a path containing all vertices of the graph.

• a cycle is a closed walk with no repeated vertices (except for v1 = vk).

• a Hamiltonian cycle is a cycle containing all vertices of the graph.

• an Eulerian cycle is a walk containing every edge of the graph exactly once.

• the between two vertices a and b, denoted dist(a, b), is the length of a shortest path joining them.

Definition A.1.5. A graph G is called connected if every pair of distinct vertices is joined by a path. Otherwise it is disconnected. A maximal (with respect to inclusion) connected subgraph of G is called a connected component of G.

Definition A.1.6. The diameter of a graph G is the maximum distance between two of G’s vertices. A disconnected graph has infinite diameter.

Definition A.1.7. Two graphs G1 = (V1,E1) and G2 = (V2,E2) are isomorphic if there is a bijection f : V1 → V2 that preserves adjacency, i.e., uv ∈ E1 if and only if f(u)f(v) ∈ E2.

Definition A.1.8. A graph G is self-complementary if G is isomorphic to its com- plement.

Definition A.1.9. In a graph,

• a is a set of pairwise adjacent vertices. The size of a maximum clique in G is called the clique number of G and is denoted ω(G).

• an independent set (also known as a stable set) is a set of pairwise non-adjacent vertices. The size of a maximum independent set in G is called the indepen- dence number (also known as the stability number) of G and is denoted α(G). 234 Appendix A. Graph Theory Background

• a matching is a set of edges no two of which share a vertex. A matching is perfect if it covers all vertices of the graph.

Definition A.1.10. A is a graph whose vertices can be divided into two disjoint sets U and V such that every edge connects a vertex in U to one in V .

Some commonly used graphs:

• Kn is the with n vertices, i.e. the graph with n vertices every two of which are adjacent.

• Kn,m is the , i.e. a bipartite graph with parts U and V of size n and m, respectively, such that each vertex in U is connected to every vertex in V .

• On is the edgeless graph on n vertices, i.e. the graph on n vertices that has no edges.

• Pn is a chordless path with n vertices, i.e. V (Pn) = {v1, v2, . . . , vn} and E(Pn) = {v1v2, . . . , vn−1vn}.

• Cn is a chordless cycle with n vertices, i.e. V (Cn) = {v1, v2, . . . , vn} and E(Cn) = {v1v2, . . . , vn−1vn, vnv1}.

• Wn is a wheel, i.e. a graph obtained from a cycle Cn by adding a vertex which is adjacent to every vertex of the cycle.

• G + H is the union of two disjoint graphs G and H, i.e. if V (G) ∩ V (H) = ∅, then V (G + H) = V (G) ∪V (H) and E(G + H) = E(G) ∪E(H). In particular, nG denotes the disjoint union of n copies of G.

Definition A.1.11. A or star Sn is the complete bipartite graph K1,n where n ≥ 0 and n = 0 corresponds to the single-vertex graph K1. The centrum of a star is the all-adjacent vertex in it.

Definition A.1.12. The Ramsey number R(n, m) gives the solution to the party problem, which asks the minimum number of guests that must be invited so that at least m will know each other or at least n will not know each other. In the language of graph theory, the Ramsey number is the minimum number of vertices v = R(n, m) such that all undirected simple graphs of order v contain a clique of order n or an independent set of order m.

The well known Ramsey’s Theorem states that such a number exists for all n and m. A.2 Directed Graphs 235 A.2 Directed Graphs

In this section, we present some terminology and notation for directed graphs. We repeat that in a directed graph every edge is an ordered pair of vertices. Frequently, directed edges are called arcs. Below, we represent an arc from a to b by (a, b), although in the book, we also use the notation a → b. Graphically, the orientation of an arc is indicated by an arrow (see Figure A.2). Observe that two vertices of an oriented graph can be connected by two arcs directed opposite to each other.

1- 2 3- 4 5

Figure A.2: An oriented graph G = (V,E) with V = {1, 2, 3, 4, 5} and E = {(1, 2), (3, 2), (3, 4), (5, 4)}

Definition A.2.1. Let G be a directed graph and v a vertex of G.

• The in-degree of v is the number of edges directed to v, i.e. of the form (u, v).

• The out-degree of v is the number of edges directed out of v, i.e. of the form (v, u).

• G is balanced if the in-degree equals the out-degree for each vertex of the graph.

Definition A.2.2. A directed graph is strongly connected if every vertex of the graph is connected to every other vertex of the graph by a directed path (i.e. a path that respects the orientation of the edges). A directed graph is weakly connected if the underlying graph is connected.

A.3 Some Graph Theory Results Used in the Book

In this section we provide two well known results in graph theory.

Theorem A.3.1. A graph contains an Eulerian cycle if and only if the graph is connected and the degree of each of its vertices is even.

Theorem A.3.2. A digraph contains an Eulerian cycle if and only if it is balanced and strongly connected. Appendix B

Non-graph-theoretical Background

Here we provide definitions of some basic notions used in the book.

B.1 Algebra and Analysis

Definition B.1.1. For sets A and B, the Cartesian product A × B is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B. That is,

A × B = {(a, b) | a ∈ A and b ∈ B}.

Definition B.1.2. Sn denotes the set of all permutations of length n. A permutation of length n is normally thought of by us as a word of length n, without repeated letters, over an n-letter alphabet, typically {1, 2, . . . , n}.

Definition B.1.3. A (non-strict) partial order is a binary relation “≤” over a set P which is reflexive, antisymmetric and transitive, i.e. which satisfies for all a, b and c in P the following conditions:

• a ≤ a (reflexivity);

• if a ≤ b and b ≤ a then a = b (antisymmetry);

• if a ≤ b and b ≤ c then a ≤ c (transitivity).

Definition B.1.4. A set with a partial order is called a partially ordered set, or a poset.

Definition B.1.5. For a, b, elements of a partially ordered set P , if a ≤ b or b ≤ a, then a and b are comparable. Otherwise they are incomparable.

© Springer International Publishing Switzerland 2015 237 S. Kitaev, V. Lozin, Words and Graphs, Monographs in Theoretical Computer Science. An EATCS Series, DOI 10.1007/978-3-319-25859-1 238 Appendix B. Non-graph-theoretical Background

Definition B.1.6. A partial order under which every pair of elements is comparable is called a total order or linear order. A totally ordered set is also called a chain. Definition B.1.7. An element a is said to be covered by another element b if a is strictly less than b and no third element c fits between them. That is, both a ≤ b and a ≠ b are true, and a ≤ c ≤ b is false for each c with a ≠ c ≠ b. Definition B.1.8. A linear extension of a partial order is a linear order that is compatible with the partial order. Definition B.1.9. The dimension of a poset P is the least integer t for which there exists a family of t linear extensions of P so that, for every x and y in P , x precedes y in P if and only if it precedes y in each of the linear extensions.

Definition B.1.10. A family R = (≤1,..., ≤t) of linear orders on a set X is called a realizer of a poset P = (X, ≤P ) if ≤p= ∩R which is to say that for any x and y in X, x ≤P y precisely when x ≤1 y, x ≤2 y, . . ., and x ≤t y. Definition B.1.11. Suppose that S is a set and · is some binary operation S ×S → S, then S with · is a monoid if it satisfies the following two axioms:

• Associativity. For all a, b and c in S, the equation (a · b) · c = a · (b · c) holds. • Identity element. There exists an elenent e in S such that for every element a in S, the equations e · a = a · e = a hold. Definition B.1.12. For a set A, the set of all words over A, denoted A∗, is called the free monoid. Definition B.1.13. Given a set A, the free commutative monoid on A, denoted [A], is the set of all finite multisets with elements drawn from A, with the monoid operation being multiset sum and the monoid unit being the empty multiset.

A morphism is a structure-preserving map from one structure to another. A special case of a morphism is defined next.

Definition B.1.14. A map f : X → Y is called an epimorphism if g1 · f = g2 · f implies g1 = g2 for all morphisms g1, g2 : Y → Z.

In the following definition, one may think of a formal power series as a power series in which we ignore questions of convergence by not assuming that the variable denotes any numerical value. Definition B.1.15. A generating function is a formal power series in one indeter- minate, whose coefficients encode information about a sequence of numbers an that is indexed by the natural numbers. The∑ordinary generating function, or just the ∞ n generating function, of a sequence an is n=0 anx . B.2 239

Generating functions are a very powerful tool in modern , in par- ticular in combinatorics. We end this section by presenting three members of the family of Bachmann- Landau notations used in this book. Definition B.1.16 (Big-O notation). f(x) = O(g(x)) means that there exists a positive real number M and a real number x0 such that |f(x)| ≤ M · |g(x)| for all x ≥ x0. Definition B.1.17 (Small-O notation). f(x) = o(g(x)) means that for all c > 0 there exists some k > 0 such that 0 ≤ f(n) < c · g(n) for all n ≥ k. The value of k must not depend on n, but may depend on c. Definition B.1.18 (Big-Theta notation). f(n) = Θ(g(n)) if f is bounded both above and below by g asymptotically. That is, there exist positive k1, k2 and n0 such that k1 · g(n) ≤ f(n) ≤ k2 · g(n) for all n > n0.

B.2 Combinatorics

Definition B.2.1. The sequence F (n) of Fibonacci numbers is defined by the re- currence relation F (n) = F (n − 1) + F (n − 2) with F (0) = 0 and F (1) = 1.

Fibonacci numbers, which have numerous applications, are probably the best known, and definitely the oldest known (dating back to ca. 200BC) sequence of num- bers. The first few Fibonacci numbers, for n = 0, 1, 2,..., are

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,.... ( ) 1 2n Definition B.2.2. The nth Catalan number is n+1 n .

The Catalan numbers occur in many counting problems, often involving re- cursively defined structures. The first few Catalan numbers, for n = 0, 1, 2,..., are

1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900,....

Definition B.2.3. A Stirling number of the second kind is the number of ways to {partition} a set of n objects into k non-empty subsets and is denoted by S(n, k) or n k .

The initial values for the Stirling numbers of the second kind are presented in Table B.1. 240 Appendix B. Non-graph-theoretical Background

n\k 0 1 2 3 4 5 6 7 0 1 1 0 1 2 0 1 1 3 0 1 3 1 4 0 1 7 6 1 5 0 1 15 25 10 1 6 0 1 31 90 65 15 1 7 0 1 63 301 350 140 21 1 Table B.1: The initial values for the Stirling numbers of the second kind

Definition B.2.4. The nth Bell number Bn counts the number of different ways to partition a set with n elements.

The first few Bell numbers, for n = 0, 1, 2,..., are

1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975,....

The following proposition is known as the pigeonhole principle. Despite being trivial, the principle has found many applications in various problems, e.g. in proving that the maximum number of non-attacking knights on the chessboard is 32.

Proposition B.2.5 (The pigeonhole principle). If n items are put into m contain- ers, with n > m, then at least one container must contain more than one item. Bibliography

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0/±1 matrix, 196, 198 4-chromatic graph, 71 1-colourable graph, 68 4-clique, 46 1-uniform word, 34, 35, 37, 39, 60, 106 4-colourable and K4-free graph, 46 1-word-representable graph, 35, 62, 83, 4-colourable graph, 46, 70, 111 99 4-cycle, 70–72, 75, 110 11-occurrence representable graph, 178 5-cycle, 35, 95 11-representable graph, viii, 131, 132, 134, 5-degenerate graph, 69 157 6-cycle, 54 12-occurrence representable graph, 178 H-graph, 196 12-representable graph, viii, 131, 132, 140, K3,3, 162 141, 143–146, 148–152, 154–158, U-cycle, 219, 220 178 U-cycle for 12-representable labelled graph, 132, 142 multi-patterns, 220 132-avoiding permutation, 184 partitions, 219 1k-representable graph, 150 permutations, 219 2-colourable graph, 68 subsets of a finite set, 219 2-connected graph, 81, 82 word patterns, 220 2-degenerate graph, 69 words, 219 2-gon, 207 χ-boundedness, 20 2-good orientation, 66 k-colourable graph, 17, 18, 68, 70 2-uniform word, 35, 60, 61, 68, 69, 78, 85 k-degenerate graph, 69 2-word-representable graph, 4, 35, 41, 68, k-generalized Fibonacci number, 222 70, 81–85 k-gon, 207 3-colourable graph, vii, 4, 49, 68–71, 77, k-good orientation, 66 94, 110–112, 117, 118, 129, 165 k-letter graph, 189–192, 197, 199 3-cycle, 72 k-letter , 196 3-dimensional cube, 88 k-polygon-circle graph, 207 3-regular graph, 69 k-regular graph, 7 3-subdivision of a graph, 87 k-subdivision of a graph, 87 3-uniform word, 35, 49, 85, 86, 89 k-uniform word, 34, 35, 37–39, 97, 98 3-word-representable graph, 35, 40, 71, k-word-representability, 38, 54, 165 84, 85, 87, 88, 128, 162, 169 k-word-representable graph, 4, 34, 35, 37– 3-word-representation, 55 39, 41, 43, 63, 65, 67, 81, 97, 98,

© Springer International Publishing Switzerland 2015 253 S. Kitaev, V. Lozin, Words and Graphs, Monographs in Theoretical Computer Science. An EATCS Series, DOI 10.1007/978-3-319-25859-1 254 Index

162 asymptotically optimal coding for a graph n-cube, 214–216, 229 class, 25 n-cycle, 35, 37, 41, 82 autocorrelation of a word, 223 n-gon, 82 Avgustinovich, S., 165 n-universal graph, 201 n-universal , 189 Bachmann-Landau notation, 239 u-representable graph, 131, 132, 134–137, balanced digraph, 209, 235 139, 157, 180, 184 base point, 198 Baudot, E.,´ 213 (bio)chemistry, 213 Bell number, xi, 203, 204, 240 Ulfarsson,´ H., 49 Berge’s Strong Perfect Graph Conjecture, u-representable graph, 177 19 Berge’s Weak Perfect Graph Conjecture, Abrham, J., 166 19 acyclic digraph, 58–60 Berge, C., 19 acyclic graph, 12 Bertrand-Chebyshev Theorem, 24 acyclic orientation, 57–59, 206, 207 bidirectional edge, 210 bijection, 186, 189, 197, 199 , 23, 26, 62, 221 binary relation, 190 Akg¨un, O,¨ ix, 47 binary representation of a number, 25 Akrobotu, P., 49 binary sequence, 189 Alekseev, V. E., 22, 24 binary word, 189 Alekseev-Bollob´as-ThomasonTheorem, 22 biology, 213 algebra, 4, 51 bipartite graph, 7, 11, 12, 14, 16, 20, 21, all-adjacent vertex, 41, 44, 45, 47, 63, 64, 23, 29, 30, 41, 68, 165, 169, 190, 67, 146 202, 234 almost periodic word, 203, 204 bipartite permutation graph, 202 alphabet, xi, 2, 6, 22, 23, 28, 31, 49, 51– Bollob´as,B., 22 53, 163, 175, 182, 189, 190, 199, Brooks’ Theorem, 18, 69 204, 205, 208, 209, 216–218, 237 bull-free graph, 20 alternating letters, 31, 35, 38, 42, 51, 59, Burstein, A., 220 64, 207 alternating sets, 49 Cameron, P., 227 alternation graph, 2 canonical coding of a graph, 23, 26, 27 alternation matrix, 54 canonical monoid epimorphism, 205 alternation word digraph, xi, 48–50, 52, canonical representation of a graph, 23 53 cardinal product of graphs, 167 antichain, 191, 205, 206 Cartesian product of graphs, 4, 95, 101, antisymmetric relation, 191 102, 129, 167, 168 asymptotic enumeration, 94 Cartesian product of sets, 237 asymptotically optimal n-universal graph, Catalan number, 239 202 categorical product of graphs, 167 Index 255

Cayley’s formula, vii, 22, 188 class cell, 198–201 (C4, 2K2,C5)-free, 12, 13 cell alphabet, 199 (C4,C5,C6,...)-free, 12 chain, 191, 238 (P4,C4)-free bipartite, 29 chain graph, 28, 190 (P 3,K3)-free, 12 Chalermsook, P., 65 2K2-free, 21 characteristic vector of an independent 2K2-free bipartite, 29 set, 188 4-colourable, 46 characterization F ree(C3,C5,C7,...), 11, 19 of 11-occurrence representable graphs, F ree(C4, 2K2,C5), 13 178 F ree(K1×K3,K1×K3,K1×P3,K1× of k-letter graphs, 190 P 3), 18 of bipartite graphs, 11 F ree(K2), 11, 19 of complete bipartite graphs, 12 F ree(P3, 2K2), 21 of cool ordering, 206 F ree(P3,K3), 18, 21 of graphs with representation num- F ree(C4, C5, C6,...), 12 ber 1, 81 F ree(K2), 11 of graphs with representation num- K3-free, 11, 12, 18 ber 2, 83 P3-free, 11, 18, 21 of hereditary classes, 7, 9 P4-free, 30, 195 of interval graphs, 16 P4-free bipartite, 30 of line graphs, 171 R1, 81, 94 of perfectly orderable graphs, 57 R2, 81, 83 of planar graphs, 16, 162 R3, 84, 92–94 of threshold graphs, 188 P 3-free, 11, 12 of word-digraphs, 208 of all graphs, 202 of word-representable graphs, 2, 49, of co-graphs, 229 61, 162 classical pattern, 176 Chen, H. Z. Q., ix, 47, 48, 166, 170 claw graph, 29, 47, 108, 109, 129, 170 , 23 claw-free graph, 20 , 12–14, 20, 46, 57, 153 clique, 11, 13, 21, 43, 67, 95, 96, 98, 99, chordless cycle, 13, 191 105, 107–109, 188, 190, 192, 199, chordless path, xii, 11, 186 200, 233, 234 chromatic number, 18–21, 68–71, 93 clique covering, 67 Chudnovsky, M., 20 Clique Covering problem, 66, 67, 77 Chung, F. R. K., 219, 220 clique number, 19–21, 233 Chv´atal,V., 188 clique-width of a graph, xi, 227–229 circle, 207 co-bipartite graph, 46 circle bipartite graph, 17 co-comparability graph, 14, 150 circle graph, v, 4, 15, 17, 28, 40, 83, 84, co-graph, 28 93, 129, 162, 168, 207, 211 co-interval graph, 131, 151, 152, 154 256 Index coding of graphs, 23, 28 cool ordering, 206 coefficient of compressibility, 22 corner graph, 154–156, 158, 159 Collins, A., 49 correlation between two words, 223 column convex polyomino, 75 cover graph, 65, 66 comb graph, 150 , xii, 63, 171, 172 combinatorial Gray code, 213 cube in a word, 163 combinatorial optimization, 20 cube-free representation of a graph, 164 combinatorics on words, 5, 163, 174, 218, cut-vertex, 81, 82 220 cycle, 12, 18, 21, 28, 29, 71, 185, 186, 233, commutative semigroup, 52 234 comparability graph, v, 7, 9, 13–15, 20, , xi, 36, 44, 47, 55, 70, 108, 28, 41, 44, 50, 51, 54, 55, 57, 59, 170, 234 64, 65, 68, 100, 131, 150, 153, cyclic shift of a word, 35, 36 154, 165, 206 cyclic word, 38 comparable permutation graph, 195 complement of a graph, xi, 4, 7, 11, 12, DAG, 50 14, 15, 20, 21, 95, 129, 152, 153, de Bruijn cycle, 218, 219 195, 196, 232 de Bruijn graph, v, viii, 1, 213, 217, 219, complement of a word, xi 229 complete bipartite graph, xii, 12, 29, 30, de Bruijn sequence, 215, 218, 219, 229 41, 63, 146, 162, 234 decreasing open line segment, 196 complete graph, xii, 5, 8, 9, 11, 18, 21, decreasing subsequence in a permutation, 28, 37, 39–41, 55, 56, 62, 70, 75, 192, 193 81, 83, 87, 93, 94, 109, 110, 127, degree of a vertex, 47 136, 152, 234 degree sequence, 29 complete multipartite graph, 11 diagram of a poset, 65 complete tripartite graph, 69 Dickson’s Lemma, 205 computer memory, 185 digraph, 59, 61, 208, 209, 231 computer science, 5, 23, 191, 213 digraph path decomposition, 208 concatenation of permutations, 55 dimension of a poset, 64 conjunction of graphs, 167 direct product of graphs, 167 connected component, 11, 12, 18, 21, 29, , 50, 59 39, 81, 187, 233 directed cycle, 59, 166 connected graph, 29, 54, 185, 233 directed graph, 181, 182, 208, 211, 231, constant speed, 203 235 constraint solver, 49, 84 directed path, 61, 69, 88, 235 contiguous subword, 191 direction, 198 convex graph, 28 disconnected graph, 195 convex polyomino, 74, 75, 110, 111, 114, discrete jump, 203 115, 129 DNA, 208 convexity theory, 20 dominant vertex, 41 Index 257 , 67 finitely generated set, 205 Dominating Set problem, 66, 67, 77 forbidden induced minor, 16, 162 domination number, 67 forbidden induced subgraph, 7, 8, 10–13, domino, 117, 118, 126, 173 15, 16, 19, 20, 162, 171, 194, 203 Doran, R. W., 214 forbidden minor, 16 double caterpillar, xi, 132, 146, 148, 149, forbidden subgraph, 162 157 forbidden vertex minor, 162 double caterpillar’s spin, 146, 148–150 forest, 12, 146 doubly exponential free spectrum, 53 Fox, M., 6 Dragon curve sequence, 216 free commutative monoid, xi, 205, 238 free monoid, 205, 238 edge contraction, 4, 16, 87, 88, 95, 96, free spectrum, 49, 53 129, 210 free spectrum of a finite monoid, xi edge deletion, 16 free spectrum problem, 53 edge-perfectness, 20 full binary tree, 149 edgeless graph, xii, 9, 11, 18, 88, 152, 163, fundamental graph, 17 178, 234 empty word, 23 Gallai, T., 14 entropy of a graph class, 22–24, 28 generating function, 238 Erd˝os,P., 71, 192, 193, 227 Gent, I., ix, 47 error correction, 214 geometric grid class, 189, 192, 197 Eulerian cycle, 165, 166, 209, 218, 233, geometric grid class of a matrix, xi 235 geometric grid class of permutations, 198 evaluation, 52, 53 geometrically griddable class of permuta- Evdokimov, A., 216 tions, 197 even cycle, 131 geometrically griddable permutation class, exact match, 176 197 exact occurrence of a word, 176 Gilmore, P. C., 14 execution sequences of recurring tasks, 6 of a graph, 70, 71 exponential speed, 203 Glen, M., ix, 49, 74, 117 extremal letter, 205 Golumbic, M. C., vi factor containment relation, 205 Graham, R., 5 factor of a word, 36, 163, 170, 191, 194, graph 204, 205 2K2, 12, 13, 188 factorial class of graphs, 228, 229 3-regular, 69 factorial speed, 203 4-regular, 166 Fibonacci number, xi, 222, 239 C3, 196 finite basis for identities, 52 C4, 12, 13, 188, 196 finitely based permutation class, 194 C5, 12, 13, 44, 45, 109, 228 finitely defined class of graphs, 194 C6, 13, 54, 55 finitely generated ideal, 206 Cn, 11, 37, 44, 55, 108, 234 258 Index

H3,3, 172, 173 of overlapping permutations, xii, 213, H4,4, 173 217, 219, 220 Hk,k, 63, 79, 172, 173 Pr3, 90, 93, 94, 130 K1, 146, 234 Prn, 88–91 K1 × H, 18 with representation number 3, 84 K2, 12, 76, 92 graph theory, 4, 20, 185, 194, 206, 218 K3, 12, 18, 95 graph’s alternation number, 39 K4, 18, 32, 33, 35, 39, 75–77, 87, 110 graph-theoretical terms, 57 K4-free, 46 Gray code, v, viii, 1, 213, 229 K5, 16, 109, 162 Gray, F., 213 Kn, 37, 41, 55, 81, 93, 127, 136–138, Green, B., 227 234 grid class of permutations, 196 K1,3, 29, 47, 71, 108–110, 129 grid graph, 76, 77, 111–115, 132, 154, K1,n, 109, 146, 234 155, 157 K3,3, 16 gridding, 198 Kk,k, 63 Gr¨otzch, H., 70 K , 41, 234 n,m Halld´orsson,M. M., ix, 2, 5, 48, 62 K , 110 n Hamiltonian cycle, 218, 220, 229, 233 L(K ), 47 n Hamiltonian path, 214, 233 L(W ), 47 n Hammer, P. L., 188, 189 L , 88–90, 92, 93 n Hamming distance, 215 P , 11, 18, 109 3 hardware testing, 213 P , 30, 109, 188, 195 4 Hasse diagram, 65 P , 108 n Hegde, R., 65 W , 107, 110 4 hereditary class of graphs, 2, 4, 7, 9, 10, W , 44–46, 48, 54, 78, 87, 88, 100, 5 15, 16, 20, 21, 23, 28, 32, 35, 46, 106, 170, 174 49, 162, 193, 194, 204 W , 44, 105, 234 n hereditary class of permutations, 193 W , 46 2n+1 hereditary property, 148, 203 P 3, 12 Higman’s Lemma, 191, 205 k-colourable, 17, 18 Hoffman, A. J., 14 co-(T2), 42, 43 horizontal sink, 75, 76 of bounded arboricity, 202 horizontal source, 75, 76 of bounded vertex degree, 202 hypergraph, 175, 231 of degree at most 1, 21 of degree at most 3, 18, 69 ideal, 205 of degree at most 4, 72, 73 identity map, 205 of degree at most k, 17, 18, 29 identity permutation, 152 of diameter k, 29 in-degree of a vertex, 235 of independence number at least k, in-neighbour, 60 29 incomparable graphs, 196 Index 259 increasing open line segment, 196 Kitaev, S., v, ix, 5, 48, 49, 117, 220 increasing sequence, 204 Konovalov, A., ix, 49, 84, 85 increasing subsequence in a permutation, Kotzig, A., 166 192, 193 Kronecker product of graphs, 167 independence number, 233 Kuratowski’s Theorem, 16, 162 independent set, 11, 13, 20, 67, 68, 188, Kuratowski, K., 162 190, 192, 199, 213, 220, 222, 224, 225, 229, 233 labelled graph, 21, 23, 28, 133, 135, 148, index of a graph class, 22 155, 203 induced cycle, 153 labelled poset, 50, 54 induced minor, 16 labelled tree, 185, 186, 188 induced subgraph, 10, 16, 40, 46, 50, 57, , xii, 81, 88–90, 92, 93, 168 58, 62, 112, 115, 132, 147, 153, language, 208 174, 188–193, 195, 196, 201, 232 leaf, 85, 185–187, 228 induced subgraph characterization, 9, 10, letter graph, xi, 189, 190, 192, 199, 210 17, 28 lettericity of a graph, 190 inessential variable, 53 lexicographical product of graphs, 168 infinite almost periodic word, 204 Limouzy, V., 66 infinite antichain, 191, 195, 196, 205 , 4, 20, 21, 28, 29, 46, 47, 49, infinite graph, 232 95, 104, 105, 107, 108, 110, 129, information theory, 20 162, 170, 171 initial permutation, xii, 37–39, 56, 69 linear extension, 59, 77, 238 internal letter, 206 linear order, 41, 64, 188, 190, 206, 238 , 207 Linton, S., ix, 49, 84, 85 interval graph, 9, 12–14, 16, 28, 29, 151, local complementation, 17 154, 162 log-exponential free spectrum, 53 interval in permutation, 194 longest chordless path, 186 interval order, 154 loop, 204 involution, 21 Lov´asz,L., 19 isolated vertex, 21, 33 Lozin, V., v, ix, 49 isomorphic graphs, 233 Lozina, I., ix isomorphism, 193 Mas´arov´aZ., 49 Jackson, B., 220 matching, 20, 55, 234 Jain, K., 65 matching-representation Wilf-equivalence, Jefferson, C., ix, 47 180 mathematics, 191 K¨onig’sTheorem, 21 matrix, 54, 198, 199 K¨onig,D., 11 matrix multiplication, 51 Kelmans, A. K., 189 maximal graph without cycles, 185 Kitaev, D., ix maximal independent set, 67 Kitaev, N., ix maximum clique, 43, 61, 62, 233 260 Index

Maximum , 5, 43, 44, 66, non-trivial module, 195 67, 77, 161 non-well-quasi-order, 191 maximum independent set, 67, 233 non-word-digraph, 208 Maximum Independent Set problem, 20, non-word-representability, 49 66, 67, 77 non-word-representable graph, 2, 42–44, maximum matching, 21 46–48, 54, 55, 70, 72, 96, 109, maximum matching problem, 20 126, 129, 168 maximum representation number of a non-word-representable line graph, 47 graph, 162 NP-complete problem, 20, 65–67, 77, 161, maximum vertex degree, 18 206, 207 McKay, B., 47 NP-hard problem, 65–67, 77, 228 minimal connected graph, 185 occurrence of a pattern, 176 minimal forbidden induced minor, 16 occurrence-representation Wilf-equivalence, minimal forbidden induced subgraph, 8, 180 10, 13, 15, 18–20, 29, 162, 171, odd cycle, 16, 18, 19 194 optimal n-universal graph, 202 minimal forbidden minor, 16 optimal coding of a graph, 28 minimal forbidden subgraph, 16 optimal representation of a graph, 24 minimal forbidden vertex-minor, 17 order diagram, 65 minimal generating set, 205 order-optimal n-universal graph, 202 minimal imperfect graph, 19 orientation of a graph, 57, 59, 72 minimal non-interval graph, 13 oriented cycle, 72 minimal non-word-representable graph, 46 oriented graph, 231 minimum vertex cover, 21 out-degree of a vertex, 235 Minion, 47, 49, 84 out-neighbour, 60 minor, 16, 87, 128, 129 outer-face, 82 minor ordering of matroid, 191 , 17, 69, 82, 84, 165 module, 4, 76, 95, 100, 194 monoid, 51–53, 199 partial multiplication matrix, 198 monomial, 204–207 partial order, 16, 154, 191, 193, 237, 238 morphism, 104, 238 partially ordered set, 204, 237 multi-permutation, 151 partially oriented graph, 72 multiset, 231 partition of a set, 21, 199 music, 208 path, 18, 28, 186, 233, 234 , 47, 108, 109, 170 neighbourhood of a vertex, 42–44, 54, 86 path-scheme, v, 165, 213, 220, 221 non-12-representable graph, 132, 145, 149 pattern class of permutations, 194 non-commutative product, 51 pattern containment, 193, 195, 196 non-comparability graph, 44, 45, 47 pattern in a word, 193 non-comparable permutation, 195 pattern involvement, 193 non-convex polyomino, 112 pattern-avoidance, 210 Index 261 pattern-avoiding class of permutations, 194poset, 64, 65, 204, 237 pattern-avoiding permutation, 183 Pr¨ufercode, 22, 23, 186 Peano word, 163 Pr¨ufersequence, vii, viii, xii, 1, 185, 186, Peng, X., 167 210, 228, 229 perfect graph, 10, 19–21, 29, 46, 162 Pr¨ufer,H., vii, 185 Perfect Graph Theorem, 19 pre-order, 191 perfectly orderable graph, 57 preference order, 169 Perkins semigroup, xi, 4, 48, 49, 51, 53, Pretzel, O., 66 54 prime graph, 194, 195 Perkins, P., 52 prism, xii, 69, 88–90, 92, 93, 130, 174 permutation, 34, 37, 41, 42, 50, 56, 59, proper n-universal graph, 202 60, 64, 68, 86, 101, 153, 189, 191– proper asymptotically optimal n-universal 195, 197, 199, 200, 219, 237 graph, 202 permutation class, 194, 195, 197 proper optimal n-universal graph, 202 permutation graph, v, viii, 7, 13–15, 17, proper order-optimal n-universal graph, 20, 28, 29, 131, 151–154, 189, 192– 202 197, 199, 200, 210 protein sequence, 208 permutation pattern, 193 Pyatkin, A., ix, 5, 48, 49 permutationally k-representable graph, 40, quadratic monomial, 206 63, 64 quasi-order, 191 permutationally representable graph, 40– quasi-ordered set, 191 42, 54, 57, 68, 70, 150 permutations Ramsey number, xii, 9, 234 213-avoiding, 202 R(5, 5), 192 321-avoiding, 202 R(6, 6), 193 avoiding 2413 and 3142, 195 Ramsey number for a graph, 192 , 32, 35, 40, 49, 71, 84, 85, Ramsey’s Theorem, 192, 234 92, 129, 130, 174 realizer, 50 pigeonhole principle, 98, 240 rectangular polyomino, 117, 118, 126 Piperno, A., 48 reduced form of a word, xii , 15–17, 69, 167, 202 reduction of a graph, xii planar matroid, 17 reflected binary code, 213 polygon, 207 reflexive relation, 191 polygon-circle graph, v, viii, 5, 207, 211 regular graph, 165 polyhedral theory, 20 regular language, 197, 199 polynomial speed, 203 regular non-word-representable graph, 166 polynomial time, 206, 207 relational product of graphs, 167 polynomially solvable problem, 44, 62, 63, Remmel, J., 131 66, 67, 77, 161 representable graph, 2 polyomino, 74–76, 110, 111, 116–118, 173 representation number of a graph, xii, 38– polyomino triangulation, 49 40, 54–56, 62, 63, 65, 68, 81, 84, 262 Index

85 simple digraph, 210 representational word, 209 simple graph, 232 reverse of a word, xii, 52, 68 simple permutation, 194, 195 Riehl, M., ix sink, 58 Robertson, N., 20 six-colour theorem, 69 robotic scheduling, 5 skew ladder graph, 154, 156, 158, 159 rooted product of graphs, 4, 95, 103, 129 snake-in-the-box problem, 2, 213, 215, 229 rooted tree, 228 software testing, 213 row convex polyomino, 75 source, 58 Ruthven, I., ix spanning subgraph, 232 speed of a graph class, 21, 28 Sagan, B., 102 speed of a property, 203 Salimov, P., 49 Spencer, J., 192 Sapozhenko, A. A., 227 spider, 207 Savile Row, 47 , 12, 13, 29, 46, 202 scheduling, 5 square in a word, 163 scheduling problem, 6 square-free antichain, 206 security patrol plan, 169 square-free monomial, 206 security patrol problem, 6 square-free representation of a graph, 164 Seif, S., ix, 48 square-free set, 206, 207 self-complementary graph, 233 square-free word, 163, 170 semi-cycle, 59 stability number, 233 semi-transitive digraph, 68, 78 stable set, 67, 233 semi-transitive graph, 58, 59 standard figure of a 0/±1 matrix, 196 semi-transitive orientation, 2, 4, 33, 41, standard gridded figure of a 0/±1 matrix, 49, 57–59, 61, 66, 71, 72, 75–78, 198 88, 96, 104, 127, 128, 174 star graph, 109, 110, 234 semi-transitively oriented graph, 60, 61, star tree, xii, 146, 234 71, 78 Stirling number of the second kind, xii, semidefinite programming, 20 209, 239, 240 semigroup, 51, 52 stochastic word-based process, 208 semigroup theory, 48 Strong Perfect Graph Conjecture, 20 semilattice, 53 Strong Perfect Graph Theorem, 19, 21 sequence, 204 strongly connected digraph, 209 sequence of σ, 215, 216 strongly connected directed graph, 235 Severs, C., 49 strongly connected word-digraph, 209 Seymour, P., 20 sub-log-exponential free spectrum, 53 shortcut, 59, 60, 62, 68, 72–76, 79, 96 subcubic graph, vii, 69, 78 shortcut-free orientation, 59, 62 subdivision graph, 69 shortcutting edge, 59 subdivision of a graph, 85, 87 shortest cycle, 70 subgraph, 16, 26, 69, 120, 191, 232 Index 263 subsequence, 60, 131, 170, 194 triangular prism, 88, 94, 100 supplement of a graph, xi triangulated graph, 12 Szekeres, G., 193 triangulation, 82, 110–120, 122, 126, 129 triangulation of a polyomino, 74, 75, 173 of graphs, 167, 168 trivial cube in a word, 163 theoretical computer science, 191, 201 trivial module, 194 theory of partial orders, 64, 190 trivial square in a word, 163 theory of patterns in permutations and words, 176, 180 unbounded face, 82 theory of permutations, 192, 194 undirected graph, 69, 204 Thomas, R., 20 undirected path, 209 Thomason, A., 22 uniform double caterpillar, 148 three-dimensional cube, 66, 88, 171 uniform word, 54, 78, 106 threshold graph, 28, 188, 189, 202, 210 uniform word-representation, 39, 175 Thue, A., 163 unit interval graph, 29 topological order, 59 universal cycle, 219 topological sort, 69 universal graph, xii, 201 topsort, 59 universal permutation, 201 total order, 238 universal point set, 202 tournament, 191 universal word, 219 Towers of Hanoi puzzle, 214 universality, 201 trace, 199 unlabelled graph, 189 trace monoid, 199 Valyuzhenich, A., 220, 226 transitive graph, 59 van Rooij, A.C.M., 108 transitive orientation, 13, 15, 55, 57, 59, variable, 51, 52 68, 128, 206 Venn diagram, 152 transitive relation, 191 Vertex Colouring problem, 66, 67, 77 transitive subgraph, 58 vertex deletion, 16 transitively orientable graph, 13, 14, 41, vertex-minor of a graph, 17 57 trapezoid graph, 28 Wagner’s conjecture, 191 tree, 7, 12, 33–35, 54, 56, 132, 141, 143, weak direct product of graphs, 167 144, 146–150, 154, 157, 158, 185, weakly chordal graph, 28 187, 210, 228 weakly connected digraph, 209, 235 triangle, 100 well-based path-scheme, 220–222, 224, 225, triangle-free graph, 11, 47, 66, 70 229 triangle-free non-word-representable graph,well-based sequence, 222, 224–227 71, 78 well-based set, 222, 226 triangle-free permutation graph, 202 well-founded partial order, 205 triangle-free planar graph, 70, 167 well-founded set, 191, 205 triangle-free word-representable graph, 66 well-quasi-order, 191 264 Index well-quasi-orderability, 189, 191, 195 well-quasi-ordered set, 189–192, 195, 197 well-quasi-ordering, xii, 190, 191 wheel graph, xii, 44–46, 48, 54, 78, 87, 105–107, 131, 150, 170, 234 Wilf, H. S., 108 Wilf-equivalence, 157, 180 Wilf-equivalent sets of patterns, 180 Winterfjord, B., 223, 224 word problem, 49, 51, 52 word-digraph, v, xi, 2, 5, 208–211 word-digraph family, xii word-graph, 208 word-representability, 49 word-representable directed graph, 59 word-representable graph, v, vii, viii, 7, 28, 31–33, 37–39, 41, 42, 44, 46, 48, 49, 51, 54, 57, 59, 61, 62, 65– 71, 75, 77, 78, 81, 88, 94, 95, 110, 127, 128, 131, 161, 163, 167, 185, 207, 210, 211 word-representant, 2, 4, 31, 33–35, 37– 39, 59, 61, 64, 105, 106, 108, 162, 163, 169 word-representation, 39 WQO, xii, 191

Yannakakis, M., 65

Zang, N., 5 Zimin word, 215, 216