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

Tiziana Calamoneri, Federico Corò, Neeldhara Misra, Giacomo Paesani

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.