Independence and global offensive alliance in graphs.
Mustapha Chellali, Lutz Volkmann · 2010
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) ≤ 3α(G) andifGis a 2 connected unicyclic graph, then γô(G) ≤ 3 α(G)+1, where α(G) isthe 2 independence number of G. Moreover, we characterize extremal bipartite graphs achieving equality in the first upper bound.