This paper investigates the concept of strong domination within zero-divisor graphs (ZDGs), aiming to compute their strong domination numbers. A zero-divisor graph (ZDG), denoted by Γ(R), is constructed by taking the nonzero zero-divisors of a commutative ring R with unity as its vertices. An edge connects two distinct vertices (u i and u j ) if their product (u i u j ) is zero. This specific definition, excluding the zero element, was a redefinition and simplification by Anderson and Livingston in 1999 from Istvan Beck's original concept in 1988, allowing for a clearer study of the properties of zero-divisors. The broader concept of domination in graphs has a notable history, initially defined by Berge 4 in 1962 and later popularised by Cockayne and Hedetniemi 5 in 1977, and remains a vibrant and rapidly developing area in graph theory. A strong dominating set (M) is a subset of ZDG vertices such that for every vertex u not in M, there exists a vertex v in M where u and v are connected by an edge (their product is zero), and crucially, the degree of u is less than or equal to the degree of v. The strong domination number (γ st (ℤ n )) is then defined as the minimum possible number of vertices in such a strong dominating set(SDS). The results related to strong domination number of Γ(ℤ n ) are stated and proved. Aside from the theoretical findings, we also formulate a Python algorithm that can calculate the strong domination number of Γ(ℤ n ).
Ali et al. (Fri,) studied this question.