Global Offensive Alliances and Groupies in Heterogeneous Random Graphs

A vertex subset S of a graph G is a global offensive alliance if every non-member v of S has at least as many neighbors inside S as outside S in the closed neighborhood of v.The global offensive alliance number γG is the cardinality of a minimal global offensive alliance. Avertex is a groupie if its degree is not less than the mean of the degrees of its neighbors. The number of groupies in G is denoted by ηG. In this paper, we study these two sort of orthogonal concepts over a heterogenous random graph G obtained by including each edge e from acomplete graph Kn ofordern withanindividual probability pn(e) independently. For a complete t-ary tree T with height 2, γT = ηT = t. In the random graph setting, it is found that γG ηG n/2undersomeneighborhood density conditions of the edge probabilities, where an bn meansan/bn → 1asn →∞