Global Defensive Alliances in the Lexicographic Product of Paths and Cycles

A set $S$ of vertices of graph $G$ is a $\mathit{\text{defensive}}\mathit{\text{alliance}}$ of $G$ if for every $v\in S$, it holds $\mathrm{\mid }N[v]\cap S\mathrm{\mid }\ge \mathrm{\mid }N[v]-S\mathrm{\mid }$. An alliance $S$ is called $global$ if it is also a dominating set. In this paper, we determine the exact values of the global defensive alliance number of lexicographic products of path and cycles.