# Eternal domination on prisms of graphs

## TL;DR

This paper investigates the eternal domination number in graphs, proving the existence of infinitely many graphs where this number equals the clique cover number, but differs when considering certain graph products.

## Contribution

It resolves a conjecture by demonstrating infinitely many graphs with specific eternal domination and clique cover properties, especially under graph Cartesian products.

## Key findings

- Existence of infinitely many graphs with ^(G)= heta(G)
- Existence of graphs where ^(G  K_2)< heta(G  K_2)
- Counterexamples to previous conjectures in graph eternal domination

## Abstract

An eternal dominating set of a graph $G$ is a set of vertices (or "guards") which dominates $G$ and which can defend any infinite series of vertex attacks, where an attack is defended by moving one guard along an edge from its current position to the attacked vertex. The size of the smallest eternal dominating set is denoted $\gamma^\infty(G)$ and is called the eternal domination number of $G$. In this paper, we answer a conjecture of Klostermeyer and Mynhardt [Discussiones Mathematicae Graph Theory, vol. 35, pp. 283-300], showing that there exist there are infinitely many graphs $G$ such that $\gamma^\infty(G)=\theta(G)$ and $\gamma^\infty(G \Box K_2)<\theta(G \Box K_2)$, where $\theta(G)$ denotes the clique cover number of $G$.