On the global offensive alliance in unicycle graphs

Abstract

For a graph G=(V,E), a set S⊆V is a dominating set if every vertex in V−S has at least a neighbor in S. A dominating set S is a global offensive alliance if for each vertex v in V−S at least half the vertices from the closed neighborhood ofv are in S. The domination number γ(G) is the minimum cardinality of a dominating set of G, and the global offensive alliance number γo(G) is the minimum cardinality of a global offensive alliance of G. We show that if G is a connected unicycle graph of order n with l(G) leaves and s(G) support vertices then γo(G)≥n−l(G)+s(G)3. Moreover, we characterize all extremal unicycle graphs attaining this bound.