Global Defensive Alliances in Graphs

Abstract

A defensive alliance in a graph $G = (V,E)$ is a set of vertices $S \subseteq V$ satisfying the condition that for every vertex $v \in S$, 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.