Some results on secure domination in zero-divisor graphs

Abstract

Let Γ(G) = (V(Γ(G)),E(Γ(G))) be a zero-divisor graph. A dominating set S of V(Γ(G)) is a secure dominating set of Γ(G) if for every vertex x ∈V(Γ(G))−S, there exists y ∈ NΓ(G) (x)∩S such that (S−{y})∪{x} is a domination set. The minimum cardinality of a secure dominating set of Γ(G) is called secure domination number. In this paper, the secure domination number of zero-divisor graphs is obtained and also studied the structure of this parameter in Γ(Zn).