On global offensive alliance in zero-divisor graphs

Abstract

Let $\G(V,E)$ be a simple graph without loops nor multiple edges. A nonempty subset $S \subseteq V$ is said a {\em global offensive alliance} if every vertex $v \in V- S$ satisfies that $\d_S(v) \geq \d_{\overline{S}}(v)+1$. The {\em global offensive alliance number} $\g^o(\Gamma)$ is defined as the minimum cardinality among all global offensive alliances. Let $R$ be a finite commutative ring with identity. In this paper, we initiate the study of the global offensive alliance number of the zero-divisor graph $\G(R)$.