A graph Formula: see text is well-dominated if every minimal dominating set is a minimum dominating set. That is, Formula: see text This condition may be weakened by stipulating that only non-minimal dominating sets contain minimum dominating sets. Under this condition, Formula: see text is not equal to Formula: see text and there may exist minimal dominating sets of cardinality greater than Formula: see text Such graphs are called weakly well-dominated graphs. A study of these graphs is made in this paper.
Jabarullah et al. (Thu,) studied this question.