On graphs whose eternal vertex cover number and vertex cover number coincide

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