On well-dominated graphs

A graph is \emph{well-dominated} if all of its minimal dominating sets have the same cardinality. We prove that at least one of the factors is well-dominated if the Cartesian product of two graphs is well-dominated. In addition, we show that the Cartesian product of two connected, triangle-free graphs is well-dominated if and only if both graphs are complete graphs of order $2$. Under the assumption that at least one of the connected graphs $G$ or $H$ has no isolatable vertices, we prove that the direct product of $G$ and $H$ is well-dominated if and only if either $G=H=K_3$ or $G=K_2$ and $H$ is either the $4$-cycle or the corona of a connected graph. Furthermore, we show that the disjunctive product of two connected graphs is well-dominated if and only if one of the factors is a complete graph and the other factor has domination number at most $2$.