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Fast Winning Strategies for the Attacker in Eternal Domination

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Graph-Theoretic Concepts in Computer Science (WG 2024)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14760))

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Abstract

Dominating sets in graphs are often used to model monitoring problems, by posting guards on the vertices of the dominating set. If an (unguarded) vertex is attacked, at least one guard can then react by moving there. This yields a new set of guards, which may not be dominating anymore. A dominating set is eternal if one 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 \(\textsf{PSPACE}\)-hard, even for graph classes where finding a minimum eternal dominating set is in \(\textsf{P}\).

We then complement this result by giving polynomial time algorithms for cographs and trees, and showing a connection with treedepth 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.

This research was supported by the ANR project P-GASE (ANR-21-CE48-0001-01).

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Notes

  1. 1.

    Note that this decomposition is not necessarily unique but, for algorithmic purposes, we only need that such a decomposition tree exists and can be computed in polynomial time which is indeed the case.

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Authors and Affiliations

  1. Universite Claude Bernard Lyon 1, CNRS, INSA Lyon, LIRIS, UMR5205, 69622, Villeurbanne, France

    Guillaume Bagan, Nicolas Bousquet, Nacim Oijid & Théo Pierron

Authors
  1. Guillaume Bagan
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  2. Nicolas Bousquet
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  3. Nacim Oijid
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  4. Théo Pierron
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Correspondence to Théo Pierron .

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Editors and Affiliations

  1. Masaryk University, Brno, Czech Republic

    Daniel Kráľ

  2. University of Primorska, Koper, Slovenia

    Martin Milanič

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Bagan, G., Bousquet, N., Oijid, N., Pierron, T. (2025). Fast Winning Strategies for the Attacker in Eternal Domination. In: Kráľ, D., Milanič, M. (eds) Graph-Theoretic Concepts in Computer Science. WG 2024. Lecture Notes in Computer Science, vol 14760. Springer, Cham. https://doi.org/10.1007/978-3-031-75409-8_2

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  • DOI: https://doi.org/10.1007/978-3-031-75409-8_2

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  • Online ISBN: 978-3-031-75409-8

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