Computing m-Eternal Domination Number of Cactus Graphs in Linear Time

In m-eternal domination attacker and defender play on a graph. Initially, the defender places guards on vertices. In each round, the attacker chooses a vertex to attack. Then, the defender can move each guard to a neighboring vertex and must move a guard to the attacked vertex. The m-eternal domination number is the minimum number of guards such that the graph can be defended indefinitely. In this paper, we study the m-eternal domination number of cactus graphs. We consider two variants of the m-eternal domination number: one allows multiple guards to occupy a single vertex, the second variant requires the guards to occupy distinct vertices. We develop several tools for obtaining lower and upper bounds on these problems and we use them to obtain an algorithm which computes the minimum number of required guards of cactus graphs for both variants of the problem.