Connectivity, line-connectivity andJ-connection of the total graph

Abstract

Our aim is to prove some results concerning the connectivity, lineconnectivity and J-connection or connection modulo J of total graphs. Only finite, nonoriented graphs without loops or multiple edges will be considered. Let G be a graph whose vertex set is V(G) and whose edge set is E(G). The elements of the set V(G)wE(G) will be called the elements of G and two elements of G are said to be associated if they are either adjacent or incident. The total graph T(G) of G is a graph whose vertex set is V(G)uE(G), two vertices being joined by an edge if and only if they are associated elements of G (see [1]). As an example of a graph G and its total graph T(G) see Fig. 1. We make an obvious distinction: small rings represent point-vertices of T(G) (x is a point-vertex x ~ E(G)). T(G) contains both G and its line-graph or interchange graph [4] L(G) as disjoint subgraphs. Remember that by definition, V[L(G)] =E(G) and two vertices of L(G) are linked by an edge if and only if the corresponding edges of G are adjacent. Edges of T(G) belonging neither to G nor L(G) form what will be called the incidence-graph I(G) of G. By definition the connectivity k(G) of G is the least number of vertices whose removal disconnects G or reduces G to a single vertex; a set of k(G) vertices satisfying this condition is called a minimal separating vertex set of G. Moreover G is n-connected if and only if k(G) >= n. On the