Eternal m-security in graphs

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 $sigma_m(G) = beta(G)$.