k-STRONG DEFENSIVE ALLIANCES IN GRAPHS

Abstract

In a simple connected graph G = (V,E), a subset S of V is a defensive alliance if every vertex v ∈ S has at most one more neighbour in V − S than it has in S. The minimum cardinality of a defensive alliance in G is called the defensive alliance number of G, denoted by a(G). A k-strong defensive alliance S is a defensive alliance in G, in which removal of any set of at most k vertices does not affect its defensive property. The k-strong defensive alliance number of G is the minimum cardinality of a k-strong defensive alliance in G, denoted by ak(G). In this paper, some properties of k-strong defensive alliances are discussed and the k-strong defensive alliance numbers of some classes of graphs are obtained.