On domination numbers of zero-divisor graphs of commutative rings

Anderson, Sarah E. (Department of Mathematics, University of St. Thomas, St. Paul, MN, USA)
Axtell, Mike (Department of Mathematics, University of St. Thomas, St. Paul, MN, USA)
Kroschel, Brenda K. (Department of Mathematics, University of St. Thomas, St. Paul, MN, USA)
Stickles, Jr., Joe A. (School of Mathematics and Computational Sciences, Millikin University, Decatur, IL, USA)

Electronic Journal of Graph Theory and Applications (EJGTA)
Vol 12, No 2 (2024): Electronic Journal of Graph Theory and Applications

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.