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A note on the eternal dominating set problem

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Abstract

We consider the “all guards move” model for the eternal dominating set problem. A set of guards form a dominating set on a graph and at the beginning of each round, a vertex not in the dominating set is attacked. To defend against the attack, the guards move (each guard either passes or moves to a neighboring vertex) to form a dominating set that includes the attacked vertex. The minimum number of guards required to defend against any sequence of attacks is the “eternal domination number” of the graph. In 2005, it was conjectured [Goddard et al. (J. Combin. Math. Combin. Comput. 52:169–180, 2005)] there would be no advantage to allow multiple guards to occupy the same vertex during a round. We show this is, in fact, false. We also describe algorithms to determine the eternal domination number for both models for eternal domination and examine the related combinatorial game, which makes use of the reduced canonical form of games.

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Notes

  1. The \(O^*\)-notation used here is similar to the O-notation but hides polynomial factors in the size of an input instance. See Woeginger (2001) and Fomin and Kratsch (2010) for more on the \(O^*\)-notation and exponential-time algorithms in general.

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Acknowledgements

S. Finbow acknowledges research support from the Natural Sciences and Engineering Research Council of Canada (Grant Application 2014-06571). Serge Gaspers is the recipient of an Australian Research Council (ARC) Future Fellowship (Project Number FT140100048) and he also acknowledges support under the ARC’s Discovery Projects Funding Scheme (DP150101134). M.E. Messinger acknowledges research support from the Natural Sciences and Engineering Research Council of Canada (Grant Application 356119-2011). The authors acknowledge support from the Games and Graphs Collaborative Research Group of the Atlantic Association for Research in Mathematical Sciences.

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Authors and Affiliations

  1. St. Francis Xavier University, Antigonish, Nova Scotia, Canada

    Stephen Finbow

  2. UNSW Sydney and Data61, CSIRO, Eveleigh, Australia

    Serge Gaspers

  3. Mount Allison University, Sackville, Canada

    Margaret-Ellen Messinger

  4. Capilano University, North Vancouver, Canada

    Paul Ottaway

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  1. Stephen Finbow
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  2. Serge Gaspers
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  3. Margaret-Ellen Messinger
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  4. Paul Ottaway
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Correspondence to Margaret-Ellen Messinger.

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Finbow, S., Gaspers, S., Messinger, ME. et al. A note on the eternal dominating set problem. Int J Game Theory 47, 543–555 (2018). https://doi.org/10.1007/s00182-018-0623-0

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  • DOI: https://doi.org/10.1007/s00182-018-0623-0

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