We consider finite graphs G with vertex set V (G). For a subset S ⊆ V (G), we define by G[S] the subgraph induced by S. By n(G) = |V (G)| and δ(G) we denote the order and the minimum degree of G, respectively. Let k be a positive integer. A subset S ⊆ V (G) is a connected global offensive k-alliance of the connected graphG, ifG[S] is connected and |N(v)∩S| ≥ |N(v)−S|+k for every vertex v ∈ V (G...

Although the literature on organizational alliances and networks focuses primarily on private sector organizations, public sector organizations have long confronted pressures to manage their environments through the formation of alliances. Moreover, several organizational theorists have posited that private and public sector alliances are becoming more prevalent, particularly in the realm of re...

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.