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arXiv · 2506.02953

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

Sarah Anderson·Mike Axtell·Brenda Kroschel·Joe Stickles
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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 $\gamma_t(\Gamma(\mathbb{Z}_2 \times D)) = 2$.

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Sarah Anderson, Mike Axtell, Brenda Kroschel, Joe Stickles. 2025-06-03. Domination and Total Domination Numbers in Zero-divisor Graphs of Commutative Rings. https://arxiv.org/abs/2506.02953

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