Global Defensive Alliances in Graphs

A defensive alliance in a graph G = (V,E) is a set of vertices S ⊆ V satisfying the condition that for every vertex v ∈ S, the number of neighbors v has in S plus one (counting v) is at least as large as the number of neighbors it has in V − S. Because of such an alliance, the vertices in S, agreeing to mutually support each other, have the strength of numbers to be able to defend themselves from the vertices in V − S. A defensive alliance S is called global if it effects every vertex in V − S, that is, every vertex in V − S is adjacent to at least one member of the alliance S. Note that a global defensive alliance is a dominating set. We study global defensive alliances in graphs. ∗Research supported in part by the South African National Research Foundation and the University of Natal. the electronic journal of combinatorics 10 (2003), #R47 1