Matching properties in connected domination critical graphs

Abstract

A dominating set of vertices S of a graph G is connected if the subgraph G [ S ] is connected. Let γ c ( G ) denote the size of any smallest connected dominating set in G . A graph G is k - γ -connected-critical if γ c ( G ) = k , but if any edge e ∈ E ( G ¯ ) is added to G , then γ c ( G + e ) ⩽ k - 1 . This is a variation on the earlier concept of criticality of edge addition with respect to ordinary domination where a graph G was defined to be k -critical if the domination number of G is k , but if any edge is added to G , the domination number falls to k - 1 . A graph G is factor-critical if G - v has a perfect matching for every vertex v ∈ V ( G ) , bicritical if G - u - v has a perfect matching for every pair of distinct vertices u , v ∈ V ( G ) or, more generally, k -factor-critical if, for every set S ⊆ V ( G ) with | S | = k , the graph G - S contains a perfect matching. In two previous papers [N. Ananchuen, M.D. Plummer, Matching properties in domination critical graphs, Discrete Math. 277 (2004) 1–13; N. Ananchuen, M.D. Plummer, 3-factor-criticality in domination critical graphs, Discrete Math. 2007, to appear [3].] on ordinary (i.e., not necessarily connected) domination, the first and third authors showed that under certain assumptions regarding connectivity and minimum degree, a critical graph G with (ordinary) domination number 3 will be factor-critical (if | V ( G ) | is odd), bicritical (if | V ( G ) | is even) or 3-factor-critical (again if | V ( G ) | is odd). Analogous theorems for connected domination are presented here. Although domination and connected domination are similar in some ways, we will point out some interesting differences between our new results for the case of connected domination and the results in [N. Ananchuen, M.D. Plummer, Matching properties in domination critical graphs, Discrete Math. 277 (2004) 1–13; N. Ananchuen, M.D. Plummer, 3-factor-criticality in domination critical graphs, Discrete Math. 2007, to appear [3].].