On Graphs with Minimal Eternal Vertex Cover Number

Abstract

The eternal vertex cover problem is a variant of the classical vertex cover problem where a set of guards on the vertices have to be dynamically reconfigured from one vertex cover to another in every round of an attacker-defender game. The minimum number of guards required to protect a graph from an infinite sequence of attacks is the eternal vertex cover number (evc) of the graph. It is known that, given a graph G and an integer k, checking whether ${\text {evc}}(G) \le k$ is NP-Hard. However, for any graph G, ${\text {mvc}}(G) \le {\text {evc}}(G) \le 2 {\text {mvc}}(G)$, where ${\text {mvc}}(G)$ is the minimum vertex cover number of G. Precise value of eternal vertex cover number is known only for certain very basic graph classes like trees, cycles and grids. Though a characterization is known for graphs for which ${\text {evc}}(G) = 2{\text {mvc}}(G)$, a characterization of graphs for which ${\text {evc}}(G) = {\text {mvc}}(G)$ remained open. Here, we achieve such a characterization for a class of graphs that includes chordal graphs and internally triangulated planar graphs. For some graph classes including biconnected chordal graphs, our characterization leads to a polynomial time algorithm to precisely determine ${\text {evc}}(G)$ and to determine a safe strategy of guard movement in each round of the game with ${\text {evc}}(G)$ guards.