On the domination and signed domination numbers of zero-divisor graph

Let R be a commutative ring (with 1) and let Z(R) be its set of zero-divisors. The zero-divisor graph Γ(R) has vertex set Z∗(R) = Z(R) \ {0} and for distinct x, y ∈ Z∗(R), the vertices x and y are adjacent if and only if xy = 0. In this paper, we consider the domination number and signed domination number on zero-divisor graph Γ(R) of commutative ring R such that for every 0 6= x ∈ Z∗(R), x 6= 0. We characterize Γ(R) whose γ(Γ(R)) + γ(Γ(R)) ∈ {n + 1, n, n − 1}, where |Z∗(R)| = n.