Matching properties in domination critical graphs Original Research Article

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 under certain assumptions regarding connectivity and minimum degree, a 3-γ-critical graph G will be either factor-critical (if |V(G)| is odd) or bicritical (if |V(G)| is even).