Eternal dominating sets on digraphs and orientations of graphs

Eternal domination is a problem that asks the following question: Can we eternally defend graph ? The principle to against an attacked vertex, changes every turn, by moving guards along edges of graph. In classical version game (Burger et al., 2004), only one guard can move at time, but in m-eternal (Goddard 2005), any number single turn. This led introduction two parameters: eternal and numbers graph, are minimum necessary infinite sequence attacks. this paper, first study (m-)eternal on digraphs. We show lot results proven for undirected graphs be generalized Then, introduce oriented domination, consists finding orientation minimizes number. prove computing coNP-hard problem, give complete characterization which 2. also these parameters trees, cycles, graphs, bipartite different kinds grids products graphs.