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A graph G is said to be k-γ-critical if the size of any minimum dominating set of vertices is k, but if any edge is added to G the resulting graph can be dominated with k-1 vertices. A graph G is factor-critical if G-v has a perfect matching for every vertex v V(G) and is bicritical if G-u-v has a perfect matching for every pair of distinct vertices u,v V(G). In the present paper, it is shown that...
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. A graph G is k-γ-connected-critical if γc(G)=k, but if any edge e∈E(G¯) is added to G, then γc(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...
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