Global offensive alliances in graphs and random graphs

A global o ensive alliance in a graph G = (V; E) is a subset S of V such that for every vertex v not in S at least half of the vertices in the closed neighborhood of v are in S. The cardinality of a minimum size global o ensive alliance in G is called the global o ensive alliance number of G. We give an upper bound on the global (strong) o ensive alliance number of a graph in terms of its degree sequence. We also study global o ensive alliances of random graphs. In particular, it is proved that if p(log n)1/2 →∞ then with high probability G(n; p) has a global o ensive alliance of size at most cn if c > 1=2 and no global o ensive alliance of size at most cn if c < 1=2.