A dominating set of a graph G(V, E) is a set of vertices D\subseteq V such that every vertex in V\D has a neighbor in D. An eternal dominating set extends this concept by placing mobile guards on the vertices of D. In response to an infinite sequence of attacks on unoccupied vertices, a guard can move to the attacked vertex from an adjacent position, ensuring that the new guards configuration remains a dominating set. In the one (all) guard(s) move model, only one (multiple) guard(s) moves(may move) per attack. The set of vertices representing the initial configuration of guards in one(all) guard move model is the eternal dominating set (m-eternal dominating set) of G. The minimum size of such a set in one(all) guard move model is called the eternal domination number (m-eternal domination number) of G, respectively. Given a graph G and an integer k, the m-Eternal Dominating Set asks whether G has an m-eternal dominating set of size at most k. In this work, we focus mainly on the computational complexity of m-Eternal Dominating Set in subclasses of chordal graphs. For split graphs, we show a dichotomy result by first designing a polynomial-time algorithm for K1,t-free split graphs with t\le 4, and then proving that the problem becomes NP-complete for t\ge 5. We showed that the problem is NP-hard on undirected path graphs. Moreover, we exhibit the computational complexity difference between the variants by showing the existence of two graph classes such that, in one, both Dominating Set and m-Eternal Dominating Set are solvable in polynomial time while Eternal Dominating Set is NP-hard, whereas in the other, Eternal Dominating Set is solvable in polynomial time and both Dominating Set and m-Eternal Dominating Set are NP-hard. Finally, we present a graph class where Dominating Set is NP-hard, but m-Eternal Dominating Set is efficiently solvable.

翻译：图G(V, E)的控制集是顶点集D\subseteq V，使得V\D中的每个顶点在D中都有一个邻接顶点。永恒控制集通过将移动守卫放置在D的顶点上来扩展这一概念。针对未占据顶点上的无限攻击序列，守卫可以从相邻位置移动到受攻击顶点，确保新的守卫配置仍构成控制集。在单守卫（全守卫）移动模型中，每次攻击仅允许一个（多个）守卫移动。单（全）守卫移动模型中表示守卫初始配置的顶点集称为G的永恒控制集（m-永恒控制集）。这两种模型下此类集合的最小规模分别称为G的永恒控制数（m-永恒控制数）。给定图G和整数k，m-永恒控制集问题需要判定G是否存在规模至多为k的m-永恒控制集。本文主要研究弦图子类中m-永恒控制集问题的计算复杂性。对于分裂图，我们通过首先为t≤4的K1,t-free分裂图设计多项式时间算法，然后证明当t≥5时该问题变为NP完全，从而展示了二分性结果。我们证明了该问题在无向路径图上是NP难的。此外，我们通过展示两类图的存在来揭示变体间的计算复杂性差异：在第一类图中，控制集和m-永恒控制集均可在多项式时间内求解，而永恒控制集是NP难的；在第二类图中，永恒控制集可在多项式时间内求解，而控制集和m-永恒控制集都是NP难的。最后，我们提出了一类控制集为NP难但m-永恒控制集可高效求解的图类。