On the structure of graphs vertex critical with respect to connected domination

Abstract

A dominating set of vertices S of a graph G is connected if the subgraph G[S] is connected. Let γc (G) denote the size of any smallest connected dominating set in G. Graph G is k-γ-connected-vertex-critical (abbreviated "kcvc") if, but if any vertex v is deleted from G, then γc (G-v)≤k-1. This concept of vertex criticality stands in contrast to the concept of criticality with respect to edge addition in which a graph G is defined to be k-connected-critical if the connected domination number of G is k, but if any edge is added to G, the connected domination number falls to k-1. It is well-known that the only 1cvc graph is K1 and the 2cvc graphs are obtained from the even complete graphs K2n, with n≥2, by deleting a perfect matching. In this paper we survey some recent results for the case when γc =3. In Sect. 14.2 we present some recently derived basic properties of 3cvc graphs, especially with respect to connectivity, and then present three new infinite families of 3cvc graphs. In Sect. 14.3, we present some new matching results for 3cvc graphs. © 2009 Springer Berlin Heidelberg.