Eternal domination on prisms of graphs

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 $G$.