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Eternal Distance-k Domination on Graphs

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Abstract

Eternal domination is a dynamic process by which a graph is protected from an infinite sequence of vertex intrusions. In eternal distance-k domination, guards initially occupy the vertices of a distance-k dominating set. After a vertex is attacked, guards “defend” by each moving up to distance k to form a distance-k dominating set, such that some guard occupies the attacked vertex. The eternal distance-k domination number of a graph is the minimum number of guards needed to defend against any sequence of attacks. The process is well-studied for the situation where \(k=1\). We introduce eternal distance-k domination for \(k > 1\). Determining whether a given set is an eternal distance-k domination set is in EXP, and in this paper we provide a number of results for paths and cycles, and relate this parameter to graph powers and domination in general. For trees we use decomposition arguments to bound the eternal distance-k domination numbers, and solve the problem entirely in the case of perfect m-ary trees.

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Acknowledgements

M.E. Messinger acknowledges research support from NSERC (grant application 2018-04059). D. Cox acknowledges research support from NSERC (2017-04401) and Mount Saint Vincent University. E. Meger acknowledges research support from Université du Québec à Montréal and Mount Allison University.The authors also thank the reviewers for the detailed reviews and suggestions for notational changes that improved the readability and quality of the results.

Funding

M.E. Messinger acknowledges research support from NSERC (grant application 2018-04059). D. Cox acknowledges research support from NSERC (2017-04401) and Mount Saint Vincent University. E. Meger acknowledges research support from Université du Québec à Montréal and Mount Allison University.

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Authors and Affiliations

  1. Laboratoire d’algèbre, du combinatoire et d’informatique mathématique, Université du Québec à Montréal, Montréal, Canada

    E. Meger

  2. Department of Mathematics and Statistics, Mount Saint Vincent University, Halifax, Canada

    D. Cox

  3. Department of Mathematics and Computer Science, Mount Allison University, Sackville, Canada

    M. E. Messinger

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  1. D. Cox
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  2. E. Meger
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Correspondence to E. Meger.

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Cox, D., Meger, E. & Messinger, M.E. Eternal Distance-k Domination on Graphs. La Matematica 2, 283–302 (2023). https://doi.org/10.1007/s44007-023-00044-3

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  • DOI: https://doi.org/10.1007/s44007-023-00044-3

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