In the emph{Eternal Domination} game, a team of guard tokens initially occupies a dominating set on a graph $G$. A rioter then picks a node without a guard on it and attacks it. The guards defend against the attack: one of them has to move to the attacked node, while each remaining one can choose to move to one of his neighboring nodes. The new guards' placement must again be dominating. This attack-defend procedure continues perpetually. The guards win if they can eternally maintain a dominating set against any sequence of attacks, otherwise, the rioter wins. We study rectangular grids and provide the first known general upper bound for these graphs. Our novel strategy implements a square rotation principle and eternally dominates $m imes n$ grids by using approximately $rac{mn}{5}$ guards, which is asymptotically optimal even for ordinary domination.
展开▼
机译：在 emph {Eternal Domination}游戏中，一组守卫令牌最初占据图形$ G $上的主导集。然后，暴徒挑起一个没有护卫的节点并对其进行攻击。守卫们防御攻击：他们之一必须移到被攻击的节点，而其余每个人都可以选择移到他的相邻节点之一。新警卫的位置必须再次占据主导地位。该攻击防御过程将永久继续。如果警卫人员可以永久地保持对任何攻击序列的统治地位，他们就会获胜，否则，暴徒获胜。我们研究矩形网格，并为这些图提供第一个已知的一般上限。我们的新颖策略实现了正方形旋转原理，并通过使用大约$ frac {mn} {5} $个警卫来永远控制$ m times n $网格，即使对于普通的控制，这也是渐近最优的。
展开▼
