Global defensive kk-alliances in graphs

Let Γ=(V,E)Γ=(V,E) be a simple graph. For a nonempty set X⊆VX⊆V, and a vertex v∈Vv∈V, δX(v)δX(v) denotes the number of neighbors vv has in XX. A nonempty set S⊆VS⊆V is a defensive  kk-alliance   in Γ=(V,E)Γ=(V,E) if δS(v)≥δS̄(v)+k,∀v∈S. A defensive kk-alliance SS is called global if it forms a dominating set. The global defensive  kk-alliance number   of ΓΓ, denoted by γka(Γ), is the minimum cardinality of a defensive kk-alliance in ΓΓ. We study the mathematical properties of γka(Γ).