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arXiv · 1907.07910

On the m-eternal Domination Number of Cactus Graphs

Václav Blažej·Jan Matyáš Křišťan·Tomáš Valla
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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 two cycles, 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 cycles, we prove that these three numbers are equal. Moreover, we present a linear-time algorithm for computing them.

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Václav Blažej, Jan Matyáš Křišťan, Tomáš Valla. 2019-07-18. On the m-eternal Domination Number of Cactus Graphs. https://arxiv.org/abs/1907.07910

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