If each minimal dominating set in a graph is minimum set, then the called well-dominated. Since seminal paper on well-dominated graphs appeared 1988, structure of from several restricted classes has been studied. In this we give complete characterization nontrivial direct products that are We prove if strong product well-dominated, both its factors When one graph, other factor being also suffic...

Let G be a connected graph. A subset S ⊆ V (G) is strong resolving dominating set of if and for every pair vertices u, v ∈ (G), there exists vertex w such that u IG[v, w] or IG[u, w]. The smallest cardinality called the domination number G. In this paper, we characterize sets in lexicographic product graphs determine corresponding number.