Efficient attack sequences in m-eternal domination

TL;DR

This paper investigates attack strategies in the m-eternal domination game on trees, providing bounds on attack sequence lengths and an efficient method to generate winning attack sequences when defenders are under-equipped.

Contribution

It establishes that on trees, attackers can win within n moves if defenders are insufficient, and introduces an efficient procedure to find such attack sequences.

Findings

Attacker can win in at most n turns on trees with insufficient guards.

An efficient procedure to produce winning attack strategies is presented.

Provides bounds on attack sequence lengths in m-eternal domination.

Abstract

We study the m-eternal domination problem from the perspective of the attacker. For many graph classes, the minimum required number of guards to defend eternally is known. By definition, if the defender has less than the required number of guards, then there exists a sequence of attacks that ensures the attacker's victory. Little is known about such sequences of attacks, in particular, no bound on its length is known. We show that if the game is played on a tree $T$ on $n$ vertices and the defender has less than the necessary number of guards, then the attacker can win in at most $n$ turns. Furthermore, we present an efficient procedure that produces such an attacking strategy.