Bounds for the $m$-Eternal Domination Number of a Graph

Mobile guards on the vertices of a graph are used to defend the graph against an infinite sequence of attacks on vertices. A guard must move from a neighboring vertex to an attacked vertex (we assume attacks happen only at vertices containing no guard and that each vertex contains at most one guard). More than one guard is allowed to move in response to an attack. The m-eternal domination number, γ∞ m (G), of a graph G is the minimum number of guards needed to defend G against any such sequence. We show that if G is a connected graph with minimum degree at least 2 and of order n ≥ 5, then γ∞ m (G) ≤ ⌊ n−1 2 ⌋ , and this bound is tight. We also prove that if G is a cubic bipartite graph of order n, then γ∞ m (G) ≤ 7n 16 .