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 vertex 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 = f1; 2; 3; 4; 5g and edge set E = f12; 23; 34; 45g. 1 2 3 4 5 Figure A.1: Graph G = (V; E) with V = f1; 2; 3; 4; 5g and E = f12; 23; 34; 45g 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 directed graph (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. Graph Theory 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 2 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 62 E(G), then u and v are nonadjacent (not connected, non-neighbours). • The neighbourhood of a vertex v 2 V (G), denoted N(v), is the set of vertices adjacent to v, i.e. N(v) = fu 2 V (G) j vu 2 E(G)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 degree of v 2 V (G), denoted deg(v), is the number of edges incident to v. Alternatively, deg(v) = jN(v)j. Definition A.1.2. The complement of a graph G = (V; E) is a graph with vertex set V and edge set E0 such that e 2 E0 if and only if e 62 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 induced subgraph 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 Hamiltonian path 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 distance 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 2 E1 if and only if f(u)f(v) 2 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 clique 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 !(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 bipartite graph 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 complete graph with n vertices, i.e. the graph with n vertices every two of which are adjacent. • Kn;m is the complete bipartite graph, 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) = fv1; v2; : : : ; vng and E(Pn) = fv1v2; : : : ; vn−1vng. • Cn is a chordless cycle with n vertices, i.e. V (Cn) = fv1; v2; : : : ; vng and E(Cn) = fv1v2; : : : ; vn−1vn; vnv1g. • 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 star or star tree 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 = f1; 2; 3; 4; 5g and E = f(1; 2); (3; 2); (3; 4); (5; 4)g 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.