Perpetually Dominating Large Grids

Abstract

In the m-\emph{Eternal Domination} game, a team of guard tokens initially occupies a dominating set on a graph $G$. An attacker then picks a vertex without a guard on it and attacks it. The guards defend against the attack: one of them has to move to the attacked vertex, while each remaining one can choose to move to one of his neighboring vertices. The new guards' placement must again be dominating. This attack-defend procedure continues eternally. The guards win if they can eternally maintain a dominating set against any sequence of attacks, otherwise, the attacker wins. The m-\emph{eternal domination number} for a graph $G$ is the minimum amount of guards such that they win against any attacker strategy in $G$ (all guards move model). We study rectangular grids and provide the first known general upper bound on the m-eternal domination number for these graphs. Our novel strategy implements a square rotation principle and eternally dominates $m \times n$ grids by using approximately $\frac{mn}{5}$ guards, which is asymptotically optimal even for ordinary domination.