Abstract
A subset S of vertices in a graph G=(V,E) is called a global offensive alliance if for every vertex v not in S, at least half of the vertices in the closed neighborhood of v are in S. A global offensive alliance D is called a global strong offensive alliance if for every vertex v not in S, more than half of the vertices in the closed neighborhood of v are in S. The global offensive alliance number (global strong offensive alliance number, respectively) of G is the minimum cardinality of a global offensive alliance (global strong offensive alliance, respectively) in G. In this paper, we present new (probabilistic) upper bounds for the global offensive alliance number as well as the global strong offensive alliance number of a graph, improving previous bounds given in Harutyunyan (2014).
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