On Graphs whose Eternal Vertex Cover Number and Vertex Cover Number Coincide

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 $G$ from an infinite sequence of attacks is the eternal vertex cover number of $G$, denoted by $evc(G)$. It is known that, given a graph $G$ and an integer $k$, checking whether $evc(G) \le k$ is NP-hard. However, it is unknown whether this problem is in NP or not. Precise value of eternal vertex cover number is known only for certain very basic graph classes like trees, cycles and grids. For any graph $G$, it is known that $mvc(G) \le evc(G) \le 2 mvc(G)$, where $mvc(G)$ is the minimum vertex cover number of $G$. Though a characterization is known for graphs for which $evc(G) = 2 mvc(G)$, a characterization of graphs for which $evc(G) = mvc(G)$ 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 $evc(G)$ and to determine a safe strategy of guard movement in each round of the game with $evc(G)$ guards. The characterization also leads to NP-completeness results for the eternal vertex cover problem for some graph classes including biconnected internally triangulated planar graphs. To the best of our knowledge, these are the first NP-completeness results known for the problem for any graph class.