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Eternal m-security in graphs
Authors: Roushini Leely Pushpam, G Navamani
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Eternal 1-secure set of a graph $G = (V, E)$ is defined as a set $S_0 \subseteq V$ that can defend against any sequence of single-vertex attacks by means of single guard shifts along edges of $G$. That is, for any $k$ and any sequence $v_1, v_2, \dots, v_k$ of vertices, there exists a sequence of guards $u_1, u_2, \dots, u_k$ with $u_i \in S_{i-1}$ and either $u_i = v_i$ or $u_iv_i \in E$, such
that each set $S_i = (S_{i-1} - \{u_i\}) \cup \{v_i\}$ is dominating. It follows that each $S_i$ can be chosen to be an eternal 1-secure set. The {\it eternal 1-security number}, denoted by $\sigma_1(G)$, is defined as the minimum cardinality of an eternal 1-secure set. The {\it Eternal $m$-security} number $\sigma_m(G)$ is defined as the minimum number of guards to handle an arbitrary sequence of single attacks using multiple-guard shifts.
In this paper we characterize the class of trees and split graphs for which $\sigma_m(G) = \gamma(G)$. We also characterize the class of trees, unicyclic graphs and split graphs for which $\sigma_m(G) = \beta(G)$.