On domination numbers of zero-divisor graphs of commutative rings

Sarah E. Anderson, Mike Axtell, Brenda K. Kroschel, Joe A. Stickles, Jr.

Abstract


Zero-divisor graphs of a commutative ring R, denoted Γ(R), are well-represented in the literature. In this paper, we consider domination numbers of zero-divisor graphs. For reduced rings, Vatandoost and Ramezani characterized the possible graphs for Γ(R) when the sum of the domination numbers of Γ(R) and the complement of Γ(R) is n - 1, n, and n + 1, where n is the number of nonzero zero-divisors of R. We extend their results to nonreduced rings, determine which graphs are realizable as zero-divisor graphs, and provide the rings that yield these graphs.

Keywords


zero divisor graph, commutative rings, domination, total domination

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DOI: http://dx.doi.org/10.5614/ejgta.2024.12.2.2

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