New upper bound on the global defensive alliance number in trees

Abstract

A global defensive alliance in a graph G = (V,E) is a dominating set S satisfying the condition that for every vertex v ∈ S, |N[v] ∩ S| ≥ |N(v) ∩ (V - S)|. In this note, a new upper bound on the global defensive alliance number of a tree is given in terms of its order and the number of support vertices. Moreover, we characterize trees attaining this upper bound.