Powerful alliances in graphs

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