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arXiv · 2401.10584

Fast winning strategies for the attacker in eternal domination

Guillaume Bagan·Nicolas Bousquet·Nacim Oijid·Théo Pierron
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Abstract

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.

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Guillaume Bagan, Nicolas Bousquet, Nacim Oijid, Théo Pierron. 2024-07-15. Fast winning strategies for the attacker in eternal domination. https://arxiv.org/abs/2401.10584

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