Alliance free sets in Cartesian product graphs

Let G = (V,E) be a graph. For a non-empty subset of vertices S ⊆ V , and vertex v ∈ V , let δS(v) = |{u ∈ S : uv ∈ E}| denote the cardinality of the set of neighbors of v in S, and let S = V − S. Consider the following condition: δS(v) ≥ δS(v) + k, (1) which states that a vertex v has at least k more neighbors in S than it has in S. A set S ⊆ V that satisfies Condition (1) for every vertex v ∈ S is called a defensive k-alliance; for every vertex v in the open neighborhood of S is called an offensive k-alliance. A subset of vertices S ⊆ V , is a powerful k-alliance if it is both a defensive k-alliance and an offensive (k+2)-alliance. Moreover, a subset X ⊂ V is a defensive (an offensive or a powerful) k-alliance free set ifX does not contain any defensive (offensive or powerful, respectively) k-alliance. In this article