Efficient attack sequences 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.