A characterization of well-dominated Cartesian products

A graph is well-dominated if all its minimal dominating sets have the same cardinality. In this paper we prove that at least one factor of every connected, well-dominated Cartesian product is a complete graph, which then allows us to give a complete characterization of the connected, well-dominated Cartesian products if both factors have order at least $2$. In particular, we show that $G\mathrm{\square }H$ is well-dominated if and only if $G\mathrm{\square }H=P_3\mathrm{\square }{K}_3$ or $G\mathrm{\square }H=K_n\mathrm{\square }{K}_n$ for some $n\ge 2$.