Dominating sets of the comaximal and ideal-based zero-divisor graphs of commutative rings

Abstract

Let R be a commutative ring with nonzero identity, and let I be an ideal of R. The ideal-based zero-divisor graph of R, denoted by Γ_I (R), is the graph whose vertices are the set {x ∈ R \ I| xy ∈ I for some y ∈ R \ I} and two distinct vertices x and y are adjacent if and only if xy ∈ I. Define the comaximal graph of R, denoted by CG(R) to be a graph whose vertices are the elements of R, where two distinct vertices ɑ and b are adjacent if and only if Rɑ + Rb = R. A nonempty set S ⊆ V of a graph G = (V, E) is a dominating set of G if every vertex in V is either in S or is adjacent to a vertex in S. The domination number γ(G) of G is the minimum cardinality among the dominating sets of G. The main object of this paper is to study the dominating sets and domination number of Γ_I (R) and the comaximal graph CG₂(R) \ J(R) (or CG_J (R) for short) where CG₂(R) is the subgraph of CG(R) induced on the nonunit elements of R and J(R) is the Jacobson radical of R.

Keywords: Comaximal, ideal-based, zero-divisor, graph, dominating set, domination number.