(Eternal) vertex cover numbers of infinite and finite grid graphs

In the eternal vertex cover problem, mobile guards on the vertices of a graph are used to defend it against an infinite sequence of attacks on its edges by moving to neighboring vertices. The eternal vertex cover problem consists of determining the minimum number of necessary guards. Motivated by previous literature, we study the vertex cover and eternal vertex cover problems on regular grids when passing from infinite to finite versions of the same graphs, and we provide either coinciding or very tight lower and upper bounds on the number of necessary guards. To this aim, we generalize the notions of minimum vertex covers and minimum eternal vertex cover in order to be well-defined for infinite grids.