Some remarks on the Roman domination number of the complement of the zero-divisor graph of a commutative ring

Articles in Press

Document Type : Original paper

Author

Department of Mathematics, Saurashtra University, Rajkot, 360005, India

Abstract

The rings considered in this paper are commutative with identity that admit at least one nonzero zero-divisor. Let R be a ring. Let Z(R) denote the set of all zero-divisors of R. Let Z(R)* denote the set of all nonzero zero-divisors of R. Recall that the zero-divisor graph of R, is an undirected graph whose vertex set is Z(R)* and distinct vertices x and y are adjacent if and only if xy = 0. This paper aims to determine the Roman domination number of the complement of the zero-divisor graph of R with the assumption that its domination number is finite.

Keywords

Main Subjects

References

[1] D.D. Anderson and M. Naseer, Beck’s coloring of a commutative ring, J. Algebra 159 (1993), no. 2, 500–514. https://doi.org/10.1006/jabr.1993.1171
[2] D.F. Anderson, T. Asir, A. Badawi, and T.T. Chelvam, Graphs from Rings, Springer, 2021.
[3] D.F. Anderson and P.S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra 217 (1999), no. 2, 434–447.
https://doi.org/10.1006/jabr.1998.7840
[4] S.E. Anderson, M.C. Axtell, B.K. Kroschel, and J.A. Stickles Jr, On domination numbers of zero-divisor graphs of commutative rings, Electron. J. Graph Theory Appl. 12 (2024), no. 2, 169–180. https://doi.org/10.5614/ejgta.2024.12.2.2
[5] M.F. Atiyah and I.G. Macdonald, Introduction to Commutative Algebra, Addison Wesley, Reading, Massachusetts, 1969.
[6] R. Balakrishnan and K. Ranganathan, A Textbook of Graph Theory, second ed., Universitext, Springer Science and Business Media, New York, 2012.
[7] E. Bastida and R. Gilmer, Overrings and divisorial ideals of rings of the form $D + M$, Michigan Math. J. 20 (1973), no. 1, 79–95. https://doi.org/10.1307/mmj/1029001014
[8] I. Beck, Coloring of commutative rings, J. Algebra 116 (1988), no. 1, 208–226. https://doi.org/10.1016/0021-8693(88)90202-5
[9] E.J. Cockayne, P.A. Dreyer Jr, S.M. Hedetniemi, and S.T. Hedetniemi, Roman domination in graphs, Discrete Math. 278 (2004), no. 1-3, 11–22. https://doi.org/10.1016/j.disc.2003.06.004
[10] R. Gilmer, Multiplicative Ideal Theory, Marcel Dekker, New York, 1972.
[11] N. Jafari Rad, S.H. Jafari, and D.A. Mojdeh, On domination in zero-divisor graphs, Canad. Math. Bull. 56 (2013), no. 2, 407–411. https://doi.org/10.4153/CMB-2011-156-1
[12] I. Kaplansky, Commutative Rings, The University of Chicago Press, Chicago, 1974.
[13] S. Khojasteh and F. Heydari, The Roman domination number of comaximal ideal graph, J. Math. Ext. 19 (2025), no. 1, 1–13. https://doi.org/10.30495/JME.2025.3147
[14] T.G. Lucas, The diameter of a zero-divisor graph, J. Algebra 301 (2006), no. 1, 174–193. https://doi.org/10.1016/j.jalgebra.2006.01.019
[15] R. Nikandish and H.R. Maimani, Dominating sets of the annihilating-ideal graphs, Electron. Notes Discrete Math. 45 (2014), 17–22. https://doi.org/10.1016/j.endm.2013.11.005
[16] D. Saleh and N. Megahed, Dot product graphs and domination number, J. Egyptian Math. Soc. 28 (2020), 31. https://doi.org/10.1186/s42787-020-00092-6
[17] S. Visweswaran, Some remarks on the dominating sets of the annihilating-ideal graph of a commutative ring, Discuss. Math. Gen. Algebra Appl. 44 (2024), no. 2, 383–412. https://doi.org/10.7151/dmgaa.1458
[18] S. Visweswaran, On the Roman domination number of the complement of sum annihilating ideal graph of a reduced ring, Arab J. Math. Sci. (2026), 1–15. https://doi.org/10.1108/AJMS-01-2026-0003
[19] S. Visweswaran , On the Roman domination number of the zero-divisor graph of a commutative ring, Asian-Eur. J. Math. (2026), 2650085. https://doi.org/10.1142/S1793557126500853
[20] S. Visweswaran and H.D. Patel, A graph associated with the set of all nonzero annihilating ideals of a commutative ring, Discrete Math. Algorithms Appl. 6 (2014), no. 4, 1450047. https://doi.org/10.1142/S1793830914500475
[21] S. Visweswaran and P. Vadhel, On the dominating sets of the complement of the annihilating ideal graph of a commutative ring, Gulf J. Math. 4 (2016), no. 1, 27–38. https://doi.org/10.56947/gjom.v4i1.54
[22] M. Ye and T. Wu, Co-maximal ideal graphs of commutative rings, J. Algebra Appl. 11 (2012), no. 6, 1250114.
https://doi.org/10.1142/S0219498812501149
Communications in Combinatorics and Optimization

Articles in Press, Accepted Manuscript
Available Online from 30 August 2026

Files

Share

How to cite

Statistics