The eternal vertex cover problem is a variant of the classical vertex cover problem defined in terms of an infinite attacker–defender game played on a graph. In each round of the game, the defender reconfigures guards from one vertex cover to another in response to a move by the attacker. The minimum number of guards required in any winning strategy of the defender when this game is played on a graph G 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)≤k is NP-hard. Further, it is known that for any graph G, mvc(G)≤evc(G)≤2mvc(G), where mvc(G) is the vertex cover number of G. Though a characterization is known for graphs for which evc(G)=2mvc(G), a characterization of graphs for which evc(G)=mvc(G) remained as an open problem, since 2009. We achieve such a characterization for a class of graphs that includes chordal graphs and internally triangulated planar graphs. For biconnected chordal graphs, our characterization leads to a polynomial time algorithm for precisely determining evc(G) and an algorithm for determining a safe strategy for guard movement in each round of the game using only evc(G) guards. Though the eternal vertex cover problem is only known to be in PSPACE in general, it follows from our new characterization that the problem is in NP for locally connected graphs, a graph class which includes all biconnected internally triangulated planar graphs. We also provide reductions establishing NP-completeness of the problem for biconnected internally triangulated planar graphs. As far as we know, this is the first NP-completeness result known for the problem for any graph class. © 2021 Elsevier B.V.