Global Defensive Alliances in the Join, Corona and Composition of Graphs

Abstract

By a defensive alliance in a graph $G$ we mean any set $S$ of vertices in $G$ such that each vertex in $S$ is adjacent to at least as many vertices inside $S$, including the vertex itself, as outside $S$. If, in addition, we require that every vertex outside a defensive alliance $S$ is adjacent to at least one vertex in $S$, then $S$ becomes a global defensive alliance. The minimum cardinality of a global defensive alliance is the global defensive alliance number of $G$. In this paper, we determine bounds for the global defensive alliance numbers in the join, corona, and composition of graphs.