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An \emph{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 \(v_1, v_2, \ldots, v_k\) of vertices, there exists a sequence of guards \(u_1, u_2, \ldots, u_k\) 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 \emph{eternal 1-security number}, denoted by \(\sigma_1(G)\), is defined as the minimum cardinality of an eternal 1-secure set. This parameter was introduced by Burger et al. [3] using the notation \(\gamma_\infty\). The \emph{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. A suitable placement of the guards is called an \emph{eternal \(m\)-secure} set. It was observed that \(\gamma(G) \leq \sigma_m(G) \leq \beta(G)\). In this paper, we obtain specific values of \(\sigma_m(G)\) for certain classes of graphs, namely circulant graphs, generalized Petersen graphs, binary trees, and caterpillars.