Eternal Domination in Trees

Abstract

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). More than one guard is allowed to move in response to an attack. The $m$-eternal domination number is the minimum number of guards needed to defend the graph. We characterize the trees achieving several upper and lower bounds on the $m$-eternal domination number.