  If each minimal dominating set in a graph is a minimum dominating set, then
the graph is called well-dominated. Since the seminal paper on well-dominated
graphs appeared in 1988, the structure of well-dominated graphs from several
restricted classes have been studied. In this paper we give a complete
characterization of nontrivial direct products that are well-dominated. We
prove that if a strong product is well-dominated, then both of its factors are
well-dominated. When one of the factors of a strong product is a complete
graph, the other factor being well-dominated is also a sufficient condition for
the product to be well-dominated. Our main result gives a complete
characterization of well-dominated Cartesian products in which at least one of
the factors is a complete graph. In addition, we conjecture that this result is
actually a complete characterization of the class of nontrivial, well-dominated
Cartesian products.
