Eternal Domination and Clique Covering

TL;DR

This paper investigates the relationship between eternal domination and clique covering in graphs, identifying minimal examples, characterizing specific graph families, and exploring properties in special graph classes.

Contribution

It provides the first known minimal graphs where eternal domination is less than clique covering, characterizes such graphs up to 11 vertices, and introduces methods to generate infinite families with this property.

Findings

Smallest graph with eternal domination less than clique covering has 10 vertices.

Complete characterization of 10- and 11-vertex graphs with this property.

Infinite families of triangle-free and circulant graphs with the property.

Abstract

We study the relationship between the eternal domination number of a graph and its clique covering number using both large-scale computation and analytic methods. In doing so, we answer two open questions of Klostermeyer and Mynhardt. We show that the smallest graph having its eternal domination number less than its clique covering number has $10$ vertices. We determine the complete set of $10$-vertex and $11$-vertex graphs having eternal domination numbers less than their clique covering numbers. We show that the smallest triangle-free graph with this property has order $13$, as does the smallest circulant graph. We describe a method to generate an infinite family of triangle-free graphs and an infinite family of circulant graphs with eternal domination numbers less than their clique covering numbers. We also consider planar graphs and cubic graphs. Finally, we show that for any integer $k \geq 2$ there exist infinitely many graphs having domination number and eternal domination number equal to $k$ containing dominating sets which are not eternal dominating sets.