On well-dominated direct, Cartesian and strong product graphs

TL;DR

This paper characterizes when direct, Cartesian, and strong product graphs are well-dominated, revealing structural conditions and providing complete characterizations for certain classes, advancing understanding of domination properties in graph products.

Contribution

It provides a complete characterization of well-dominated direct and Cartesian product graphs, especially when factors are complete or well-dominated, and proposes a conjecture for Cartesian products.

Findings

Strong product is well-dominated only if both factors are well-dominated.

When one factor of a strong product is complete, the other being well-dominated suffices.

Complete characterization of well-dominated Cartesian products with at least one complete factor.

Abstract

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.