ON THE DEFENSIVE ALLIANCES IN GRAPH

Let G = (V;E) be a graph. We say that S c- V is a defensive alliance if for every u 2 S, the number of neighbors u has in S plus one (counting u) is at least as large as the number of neighbors it has outside S. Then, for every vertex u in a defensive alliance S, any attack on a single vertex by the neighbors of u in V -, we study alliances that are containing a given vertex u and study their mathematical properties.