On well-dominated graphs

TL;DR

This paper characterizes when various graph products are well-dominated, revealing conditions under which the minimal dominating sets are uniformly sized, with specific results for Cartesian, direct, and disjunctive products.

Contribution

It provides new characterizations of well-dominated graphs under different graph product operations, extending understanding of domination properties in graph theory.

Findings

Cartesian product is well-dominated only if one factor is well-dominated.

For triangle-free graphs, the Cartesian product is well-dominated only if both are K2.

Disjunctive product is well-dominated if one factor is complete and the other has domination number at most 2.

Abstract

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 $2$.