strong alliances in graphs

For any simple connected undirected graph g = (vɛe), a defensive al- liance is a subset s of v satisfying the condition that every vertex v 2 s has at most one more neighbour in v - s than it has in s. the minimum cardinality of any de- fensive alliance in g is called the alliance number of g, and is denoted by a_d(g). in this paper, we introduce a new type of alliance number called the k-strong alliance number and its varieties. the bounds for 1-strong alliance number in terms of different graphical parameters are determined and the characterizations of graphs with 1-strong alliance number 1, 2, and n are obtained.