A characterization of well-dominated Cartesian products

Abstract

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 $G\square H$ is well-dominated if and only if $G\square H=P_3\square K_3$ or $G\square H=K_n\square K_n$ for some $n\ge 2$.

arXiv comment17 pages