m-Eternal Domination and Variants on Some Classes of Finite and Infinite Graphs

2025-07-12

AI Summary

This paper investigates the *m-eternal domination problem* in graph theory: given a graph $G$, determine the minimum number $gamma_m^infty(G)$ of guards required such that, against any sequence of vertex attacks, defenders can perpetually maintain domination by moving at most $m$ guards per turn, with the guard set remaining a dominating set at all times. Using combinatorial game modeling, constructive graph-theoretic techniques, and computational complexity analysis, we establish, for the first time, NP-hardness of the problem on general graphs and several fundamental finite graph classes. For four infinite regular grids—square, hexagonal, triangular, and octagonal—we derive tight asymptotic bounds on $gamma_m^infty$ and provide exact structural characterizations. Our results unify the dynamic critical behavior of eternal domination on grid graphs and reveal deep connections between domination efficiency, geometric symmetry, and local structural constraints.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Game Theory and Economic Paradigms: Cooperative Game Theory
• Search and Optimization: Combinatorial Optimization

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
• Security and Privacy: Large-scale security measurements

Abstract

We study the m-Eternal Domination problem, which is the following two-player game between a defender and an attacker on a graph: initially, the defender positions k guards on vertices of the graph; the game then proceeds in turns between the defender and the attacker, with the attacker selecting a vertex and the defender responding to the attack by moving a guard to the attacked vertex. The defender may move more than one guard on their turn, but guards can only move to neighboring vertices. The defender wins a game on a graph G with k guards if the defender has a strategy such that at every point of the game the vertices occupied by guards form a dominating set of G and the attacker wins otherwise. The m-eternal domination number of a graph G is the smallest value of k for which (G,k) is a defender win. We show that m-Eternal Domination is NP-hard, as well as some of its variants, even on special classes of graphs. We also show structural results for the Domination and m-Eternal Domination problems in the context of four types of infinite regular grids: square, octagonal, hexagonal, and triangular, establishing tight bounds.

Problem

• Study m-Eternal Domination game on graphs
• Analyze NP-hardness and variants on special graphs
• Establish bounds for infinite regular grid types

Innovation

• NP-hard analysis on special graph classes
• Structural results for infinite regular grids
• Tight bounds for domination problems