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：图中的支配集通常用于建模对图的监控：守卫被部署在支配集的顶点上，当未受保护的顶点遭受攻击时，守卫可通过移动至该位置进行响应（从而形成新的守卫集，但此时该集可能不再具有支配性）。若一个支配集能无限抵抗攻击，则称其为永恒的。从攻击者的角度来看，若给定一个非永恒的支配集，问题在于确定需要多久才能引发一次邻近守卫无法处理的攻击。我们从计算复杂性的角度研究了该问题，证明即使对于最小永恒支配集可在多项式时间内求解的图类，该问题仍是PSPACE-hard的。随后，我们通过给出余图（cographs）和树（trees）的多项式时间算法来补充这一结果，并揭示了后者与树深度（tree-depth）的联系。此外，我们还从参数化复杂性视角研究了该问题，主要考虑两个参数：守卫数量与攻击步数。