Weakly well-dominated graphs

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.