Domination and Total Domination Numbers in Zero-divisor Graphs of Commutative Rings

Abstract

Zero-divisor graphs of commutative rings are well-represented in the literature. In this paper, we consider dominating sets, total dominating sets, domination numbers and total domination numbers of zero-divisor graphs. We determine the domination and total domination numbers of zero-divisor graphs are equal for all zero-divisor graphs of commutative rings except for ${\mathbb{Z}}_2\times D$ in which $D$ is a domain. In this case, $\gamma (\Gamma ({\mathbb{Z}}_2\times D))=1$ and ${\gamma }_t(\Gamma ({\mathbb{Z}}_2\times D))=2$.