Abstract
Given a graph G, guards are placed on vertices of G. Then vertices are subject to an infinite sequence of attacks so that each attack must be defended by a guard moving from a neighboring vertex. The m-eternal domination number is the minimum number of guards such that the graph can be defended indefinitely. In this paper we study the m-eternal domination number of cactus graphs, that is, connected graphs where each edge lies in at most one cycle, and we consider three variants of the m-eternal domination number: first variant allows multiple guards to occupy a single vertex, second variant does not allow it, and in the third variant additional “eviction” attacks must be defended. We provide a new upper bound for the m-eternal domination number of cactus graphs, and for a subclass of cactus graphs called Christmas cactus graphs, where each vertex lies in at most two biconnected components, we prove that these three numbers are equal. Moreover, we present a linear-time algorithm for computing them.
V. Blažej and T. Valla acknowledge the support of the OP VVV MEYS funded project CZ.02.1.01/0.0/0.0/16_019/0000765 “Research Center for Informatics”.
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References
Braga, A., de Souza, C.C., Lee, O.: The eternal dominating set problem for proper interval graphs. Inf. Process. Lett. 115(6), 582–587 (2015)
Burger, A.P., Cockayne, E.J., Gründlingh, W.R., Mynhardt, C.M., van Vuuren, J.H., Winterbach, W.: Infinite order domination in graphs. J. Comb. Math. Comb. Comput. 50, 179–194 (2004)
Finbow, S., Messinger, M.-E., van Bommel, M.F.: Eternal domination on 3 \(\times \) n grid graphs. Australas. J. Comb. 61, 156–174 (2015)
Goddard, W., Hedetniemi, S.M., Hedetniemi, S.T.: Eternal security in graphs. J. Comb. Math. Comb. Comput. 52, 169–180 (2005)
Gupta, U.I., Lee, D.-T., Leung, J.Y.-T.: Efficient algorithms for interval graphs and circular arc graphs. Networks 12(4), 459–467 (1982)
Harary, F.: Graph Theory. Addison-Wesley, Boston (1969)
Henning, M.A., Klostermeyer, W.F., MacGillivray, G.: Bounds for the m-eternal domination number of a graph. Contrib. Discrete Math. 12(2), 91–103 (2017). ISSN 1715-0868
Klostermeyer, W.F., MacGillivray, G.: Eternal dominating sets in graphs. J. Comb. Math. Comb. Comput. 68, 97–111 (2009)
Klostermeyer, W.F., Mynhardt, C.M.: Protecting a graph with mobile guards. Appl. Anal. Discrete Math. 10, 1–29 (2014)
Leighton, T., Moitra, A.: Some results on greedy embeddings in metric spaces. Discrete Comput. Geom. 44(3), 686–705 (2010)
Tarjan, R.: Depth-first search and linear graph algorithms. SIAM J. Comput. 1(2), 146–160 (1972)
van Bommel, C.M., van Bommel, M.F.: Eternal domination numbers of 5 \(\times \) n grid graphs. J. Comb. Math. Comb. Comput. 97, 83–102 (2016)
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We would like to thank Martin Balko and an anonymous referee for their valuable comments and insights.
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Blažej, V., Křišt’an, J.M., Valla, T. (2019). On the m-eternal Domination Number of Cactus Graphs. In: Filiot, E., Jungers, R., Potapov, I. (eds) Reachability Problems. RP 2019. Lecture Notes in Computer Science(), vol 11674. Springer, Cham. https://doi.org/10.1007/978-3-030-30806-3_4
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DOI: https://doi.org/10.1007/978-3-030-30806-3_4
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