Eternal domination is a dynamic process by which a graph is protected from an infinite sequence of vertex intrusions. In eternal $k$-domination, guards initially occupy the vertices of a $k$-dominating set. After a vertex is attacked, guards "defend" by each move up to distance $k$ to form a $k$-dominating set containing the attacked vertex. The eternal $k$-domination number of a graph is the minimum number of guards needed to defend against any sequence of attacks. The process is well-studied for the $k=1$ situation and we introduce eternal $k$-domination for $k > 1$. Determining if a given set is an eternal $k$-domination set is in EXP, and in this paper we provide a number of results for paths and cycles, and relate this parameter to graph powers and domination in general. For trees we utilize decomposition arguments to bound the eternal $k$-domination numbers, and solve the problem entirely in the case of perfect $m$-ary trees.

中文翻译：

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图上的永恒k控制

永恒统治是一个动态过程，可以保护图免受无限次的顶点入侵。在永恒的$ k $统治下，守卫最初占据$ k $统治集中的顶点。顶点受到攻击后，后卫每次向上移动$ k $来“防御”以形成一个包含受攻击顶点的$ k $主导集。图的永恒$ k $支配数是防御任何攻击序列所需的最少防护数量。对于$ k = 1 $的情况，对该过程进行了充分的研究，我们引入了对于$ k> 1 $的永恒$ k $支配。确定给定集合是否为永恒的$ k $支配集合在EXP中，在本文中，我们提供了许多路径和循环的结果，并将此参数与图形幂和支配关系相关联。