A tight bound on the cardinalities of maximum alliance-free and minimum alliance-cover sets

A defensive k-alliance in a graph G = (V, E) is a set of vertices A ⊆ V such that for every vertex v ∈ A, the number of neighbors v has in A is at least k more than the number of neighbors it has in V − A (k is a measure of the strength of alliance). In this paper, we deal with two types of sets associated with defensive k-alliances; maximum defensive k-alliance free and minimum defensive k-alliance cover sets. Define a set X ⊆ V to be maximum defensive k-alliance free if X does not contain any defensive kalliance and is a largest such set. A set Y ⊆ V is called minimum defensive k-alliance cover if Y contains at least one vertex from each defensive kalliance and is a set of minimum cardinality satisfying this property. We present bounds on the cardinalities of maximum defensive k-alliance free and minimum defensive k-alliance cover sets.