Dominating sets in graphs are often used to model some monitoring of the graph: guards are posted on the vertices of the dominating set, and they can thus react to attacks occurring on the unguarded vertices by moving there (yielding a new set of guards, which may not be dominating anymore). A dominating set is eternal if it can endlessly resist to attacks. From the attacker's perspective, if we are given a non-eternal dominating set, the question is to determine how fast can we provoke an attack that cannot be handled by a neighboring guard. We investigate this question from a computational complexity point of view, by showing that this question is PSPACE-hard, even for graph classes where finding a minimum eternal dominating set is in P. We then complement this result by giving polynomial time algorithms for cographs and trees, and showing a connection with tree-depth for the latter. We also investigate the problem from a parameterized complexity perspective, mainly considering two parameters: the number of guards and the number of steps.

翻译：图论中的支配集常用于建模对图的监控：守卫被部署在支配集的顶点上，从而能够对未受保护顶点上发生的攻击做出反应，通过移动至该位置进行应对（这会产生新的守卫集合，但该集合可能不再是支配集）。若一个支配集能够无休止地抵御攻击，则称其为永恒支配集。从攻击者的视角出发，若给定一个非永恒支配集，问题的关键在于确定我们能够以多快的速度引发一次相邻守卫无法处理的攻击。我们从计算复杂性的角度研究了这一问题，证明该问题属于PSPACE难问题，即使在寻找最小永恒支配集属于P类的图类中也是如此。随后，我们通过给出补图和树的多项式时间算法，并展示后者与树深度的关联，对这一结果进行了补充。我们还从参数化复杂性的角度研究了该问题，主要考虑两个参数：守卫数量和步骤数。