On the number of components of a graph

Let G := (V,E) be a simple graph; for I ⊆ V we denote by l(I) the number of components of G[I], the subgraph of G induced by I. For V1, . . . , Vn finite subsets of V , we define a function β(V1, . . . , Vn) which is expressed in terms of l `Sn i=1 Vi ́ and l(Vi ∪ Vj) for i ≤ j. If V1, . . . , Vn are pairwise disjoint finite independent subsets of V , the number β(V1, . . . , Vn) can be computed in terms of the cyclomatic numbers of G ˆSn i=1 Vi ̃ and G[Vi ∪ Vj ] for i 6= j. In the general case, we prove that β(V1, . . . , Vn) ≥ 0 and characterize when β(V1, . . . , Vn) = 0. This special case yields a formula expressing the length of members of an interval algebra [6] as well as extensions to pseudo-tree algebras. Other examples are given. 1. Presentation of the Main Result 1.1. Main Result. Let G := (V,E) be a graph, where V is the vertex set and E is the edge set. We suppose that E is a subset of the set [V ]2 of unordered pairs of V . Let I be a subset of V , we denote by G[I] the graph (I, E ∩ [I]2) induced by G on I. We denote by l(G[I]), or lG(I), or more simply l(I) if there is no ambiguity, the number of components of the graph G[I]. As much as possible, we abbreviate component of G[I] by component of I. We assume that lG(∅) = 0. Definition 1.1. To an integer n and a family (V1, . . . , Vn) of finite subsets of V we associate a number, denoted βG(V1, . . . , Vn), or β(V1, . . . , Vn) if there is no ambiguity, and defined as follows: β(V1, . . . , Vn) := l ( n ⋃