Connected global offensive k-alliances in graphs

We consider finite graphs G with vertex set V (G). For a subset S ⊆ V (G), we define by G[S] the subgraph induced by S. By n(G) = |V (G)| and δ(G) we denote the order and the minimum degree of G, respectively. Let k be a positive integer. A subset S ⊆ V (G) is a connected global offensive k-alliance of the connected graphG, ifG[S] is connected and |N(v)∩S| ≥ |N(v)−S|+k for every vertex v ∈ V (G)−S, where N(v) is the neighborhood of v. The connected global offensive k-alliance number γ o (G) is the minimum cardinality of a connected global offensive k-alliance in G. In this paper we characterize connected graphs G with γ o (G) = n(G). In the case that δ(G) ≥ k ≥ 2, we also characterize the family of connected graphs G with γ o (G) = n(G)−1. Furthermore, we present different tight bounds of γ o (G).