Some results related to the toughness of 3-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. The structure of k–γ-critical graphs remains far from completely understood, even in the special case when the domination number γ=3. In a 1983 paper, Sumner and Blitch proved a theorem which may regarded as a result related to the toughness of 3-γ-critical graphs which says that if S is any vertex cutset of such a graph, then G−S has at most |S|+1 components. In the present paper, we improve and extend this result considerably.