$m$-Eternal Dominating Set Problem on Subclasses of Chordal Graphs

2026-02-02

AI Summary

This study investigates the computational complexity of the $m$-eternal domination problem—determining whether a graph admits an $m$-eternal dominating set of size at most $k$—on subclasses of chordal graphs. By combining graph-theoretic modeling, polynomial-time algorithm design, and NP-completeness reductions, the work establishes the first complexity dichotomy on split graphs based on $K_{1,t}$-freeness: the problem is solvable in polynomial time when $t \leq 4$, but becomes NP-complete for $t \geq 5$. Furthermore, it proves that the problem remains NP-hard on undirected path graphs. Through the construction of several graph-class instances, the paper also elucidates fundamental complexity distinctions among dominating sets, eternal dominating sets, and $m$-eternal dominating sets.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Models for Web evolution

Abstract

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.

Problem

• m-Eternal Dominating Set
• Chordal Graphs
• Computational Complexity
• NP-completeness
• Graph Domination

Innovation

• m-eternal dominating set
• chordal graphs
• computational complexity
• split graphs
• dichotomy result