Independence and global offensive alliance in graphs

Let G be a simple graph with vertex set V (G). A non-empty set S ⊆ V (G) is a global strong offensive alliance if for every vertex v in V (G)−S, a strict majority of its closed neighborhood is in S. The global strong offensive alliance number γô(G) is the minimum cardinality of a global strong offensive alliance of G. We show that if G is a connected bipartite graph of order at least three, then γô(G) ≤ 32α(G) and if G is a connected unicyclic graph, then γô(G) ≤ 32α(G) + 1, where α(G) is the independence number of G. Moreover, we characterize extremal bipartite graphs achieving equality in the first upper bound.