  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 neighbor vertices. The eternal vertex cover problem consists in
determining the minimum number of necessary guards. Motivated by previous
literature, in this paper, we study the vertex cover and eternal vertex cover
problems on regular grids, when passing from infinite to finite version 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.
