On global (strong) defensive alliances in some product graphs

Abstract

A defensive alliance in a graph is a set $S$ of vertices with the property that every vertex in $S$ has at most one more‎ ‎neighbor outside of $S$ than it has inside of $S$‎. ‎A defensive alliance $S$ is called global if it forms a dominating set‎. ‎The global defensive alliance number of a graph $G$ is the minimum cardinality of a global defensive alliance in $G$‎. ‎In this article we study the global defensive alliances in Cartesian product graphs‎, ‎strong product graphs and direct product graphs‎. ‎Specifically we give several bounds for the global defensive alliance number of these graph products and express them in terms of the global defensive alliance numbers of the factor graphs.

Keywords

• Defensive alliances‎
• ‎global defensive alliances‎
• ‎domination‎
• ‎Cartesian product graphs‎
• ‎strong product graph‎
• ‎direct product graphs.