Discrete Mathematics..

488 Graph Theorytwo quite different meanings, although it is usually clear from the context whichmeaning is intended.What, then, is a ‘graph’? Intuitively, a graph is simply a collection of points,called ‘vertices’, and a collection of lines, called ‘edges’, each of which joinseither a pair of points or a single point to itself. A familiar example, which servesas a useful analogy, is a road map which shows towns as vertices and the roadsjoining them as edges.For mathematical purposes we require a more precise definition. In order to definea graph, we first need to specify the set of its vertices and the set of its edges. Thenwe need to say, in precise mathematical terms, which edges join which vertices.An edge is defined as having a vertex at each end, so we need to associate withevery edge of the graph its endpoint vertices. The endpoints of an edge are eithera pair of vertices (if the edge joins two different vertices) or a single vertex (if theedge joins a vertex to itself). Thus for every edge e of a graph we define a set{v 1 ,v 2 } of vertices which specifies that e joins vertices v 1 and v 2 , where of coursewe need to allow the possibility that v 1 = v 2 . Now this set {v 1 ,v 2 },whichwedenote by δ(e), is a subset of the set of vertices. Therefore δ(e) is an element ofthe power set of the vertex set. This leads us to the following formal definition.Its rather technical nature should not be allowed to obscure the essentially simpleconcept that is being described.An undirected graph Ɣ comprises:Definition 10.1(i)(ii)(iii)a finite non-empty set V of vertices,a finite set E of edges,anda function δ : E → È(V ) such that, for every edge e, δ(e) is aone- or two-element subset of V .The edge e is said to join the element(s) of δ(e).Generally we shall use the term ‘graph’ without qualification to mean undirectedgraph. If we need to emphasize a specific graph Ɣ we will write V Ɣ , E Ɣ and δ Ɣfor the sets V and E and the function δ : E → È(V ) respectively.As we have explained, this function δ is merely a formal way of specifyingthe ends of edges. In this case where an edge e joins a vertex to itself, the