On the m-eternal Domination Number of Cactus Graphs

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.

Computing m-Eternal Domination Number of Cactus Graphs in Linear Time

“…We remark that using reductions t 1 , t 2 , t 3 , c 1 , c 4 , and a small set of constant component reductions is sufficient to solve so-called Christmas cactus graphs (graphs where each edge is in at most one cycle and each vertex is in at most two 2-connected components) for which the optimal strategy we presented in [1]. The remaining reductions tackle vertices of color 2, which are not present in the class of Christmas cactus graphs.…”

Eternal k-domination on graphs

“…The eternal domination number of a graph, denoted γ ∞ all is the minimum number of guards required to defend against any sequence of attacks, where the subscript and superscript indicate that all guards can move in response to an attack and the sequence of attacks is infinite. Given the complexity of determining the eternal domination number of a graph for the all-guards move model, recent work such as [1,3,4,5,10,12,2], has focused primarily on bounding or determining the parameter for particular classes of graphs.…”