Maximum Alliance-Free and Minimum Alliance-Cover Sets

Abstract

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 as large as the number of neighbors it has in V−S plus k (where k is the strength of k:–alliance). An offensive k–alliance is a set of vertices A ⊆ V such that for every vertex v ∈ ∂ dA, the number of neighbors v has in A is at least as large as the number of neighbors it has in V − S plus k. In this paper, we deal with two types of sets associated with these k–alliances that we refer to as maximum k–alliance free and minimum k–alliance cover sets respectively. We define a set X ⊆ V to be maximum k–alliance free (for some type of k–alliance) if X does not contain any k–alliance (of that type) and X is a set of largest cardinality among all such sets. A set Y ⊆ V is called minimum k–alliance cover (for some type of k–alliance) if Y contains at least one vertex from each k–alliance (of that type) and is a set of minimum cardinality satisfying this property. We present bounds on the cardinality of maximum k–alliance free and minimum alliance k–cover sets and explore their inter-relation. The existence of forbidden subgraphs for graphs induced by these sets is also questioned.