Matching properties in connected domination critical graphs Original Research Article

A dominating set of vertices image of a graph image is connected if the subgraph image is connected. Let image denote the size of any smallest connected dominating set in image. A graph image is image-image-connected-critical if image, but if any edge image is added to image, then image. This is a variation on the earlier concept of criticality of edge addition with respect to ordinary domination where a graph image was defined to be image-critical if the domination number of image is image, but if any edge is added to image, the domination number falls to image. A graph image is factor-critical if image has a perfect matching for every vertex image, bicritical if image has a perfect matching for every pair of distinct vertices image or, more generally, image-factor-critical if, for every set image with image, the graph image 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 .] 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 image with (ordinary) domination number 3 will be factor-critical (if image is odd), bicritical (if image is even) or 3-factor-critical (again if image 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 .].