A Note on Defensive Alliances in Graphs

Abstract

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 it 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. We prove two conjectures posed by Hedetniemi, Hedetniemi, and Kristiansen in their introductory paper on alliances.