Matching properties in connected domination critical graphs

A dominating set of vertices S of a graph G is connected if the subgraph G vertical bar S vertical bar is connected. Let gamma(c)(G) denote the size of any smallest connected dominating set in G. A graph G is k-gamma-connected-critical if gamma(c)(G) = k, but if any edge e is an element of E((G) over bar) is added to G, then gamma(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 is an element of V(G),bicritical if G - u - v has a perfect matching for every pair of distinct vertices u, v is an element of V(G) or, more generally, k-factor-critical if, for every set S subset of V(G)with vertical bar S vertical bar = 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 vertical bar V(G)vertical bar is odd), bicritical (if vertical bar V(G)vertical bar is even) or 3-factor-critical (again if vertical bar V(G)vertical bar 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].]. 0 2007 Elsevier B.V. All rights reserved.