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 (where k is the strength of defensive k−alliance). An offensive k−alliance 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 (where ∂A is the boundary of set A and is defined as N [A] − A). In this paper, we deal with two types of sets associated with these k−alliances: maximum k−alliance free and minimum k−alliance cover sets. 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 is a largest such set. 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 cardinalities of maximum k−alliance free and minimum k−alliance cover sets and explore their inter-relation. The existence of forbidden subgraphs for graphs induced by these sets is also explored.