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.

翻译：在永久的顶部覆盖问题中,一个图形顶部的移动卫兵被用来通过移动到邻近的顶部来防范其边缘的无穷次攻击。永久顶部覆盖问题在于确定最低必要守卫人数。根据先前的文献,我们在本文中研究了顶部覆盖和永久顶部覆盖常规网格上的问题,从无限版的同一图形传递到有限的版本,我们提供了同步或非常紧凑的下限和上限的必要守卫人数。为此,我们推广了最低顶层覆盖和最低永久顶层覆盖的概念,以便明确界定无限的网格。