On the Roman domination number of the complement of sum annihilating ideal graph of a reduced ring

Purpose This paper aims to determine the Roman domination number of the complement of sum annihilating ideal graph of a reduced ring with the assumption that its domination number is finite. We have studied in a previous work the domination number of the complement of sum annihilating ideal graph of a commutative ring. As the Roman domination number is of historical importance, we wish to find out the Roman domination number of the mentioned graph in this paper. Design/methodology/approach We use techniques and methods from commutative ring theory regarding the minimal prime ideals. We use the graph-theoretic properties of the sum annihilating ideal graph of a commutative ring and its complement. Findings We have noted that if the number of minimal prime ideals of a reduced ring R that is not an integral domain is finite, then the domination number of the complement of sum annihilating ideal graph of a reduced ring is finite and in such a case, if the number of minimal prime ideals equals 2, then the Roman domination number of this graph is either 2 or 4 and we are able to characterize R such that the Roman domination number of this graph equals 2 (respectively, 4). If the number of minimal prime ideals of R equals 3, then it is determined that the Roman domination number of this graph is either 4 or 5. If the number of minimal prime ideals of R equals n and if n exceeds 4, then the Roman domination number of this graph is 2n or 2n −1. We are able to characterize R, according to the Roman domination number of the considered graph. Research limitations/implications We do not know any necessary and sufficient condition such that the domination number of the complement of sum annihilating ideal graph of a reduced ring is finite. If the number of minimal prime ideals of a reduced ring is finite, then the domination number of the considered graph is finite and in such cases, we have computed the Roman domination number of the considered graph. We have not considered non-reduced rings. The study helps to understand the structure of reduced rings with the desired Roman domination number of the considered graph. Practical implications The problem discussed in this work helps that there is an interplay between the graph parameter Roman domination number of a graph and the algebraic structure for which this graph parameter is considered. Social implications This paper will help researchers working in Algebra to understand algebraic structures by associating suitable graphs with algebraic structures and study the graph parameter Roman domination number. Researchers working in other fields of Mathematics can also try to undertake such an investigation. Originality/value The results mentioned in this paper are original. We first studied some known results proved by Cockayne et al. on the Roman domination number of a graph and i also studied similar work on comaximal ideal graphs of rings and determined the Roman domination number of the complement of sum annihilating ideal graph of a reduced ring. The work undertaken in this paper was not considered earlier by others.