Powerful alliances in graphs

For a graph G = ( V , E ) , a non-empty set S ⊆ V is a defensive alliance if for every vertex v in S , v has at most one more neighbor in V − S than it has in S , and S is an offensive alliance if for every v ∈ V − S that has a neighbor in S , v has more neighbors in S than in V − S . A powerful alliance is both defensive and offensive. We initiate the study of powerful alliances in graphs.