# 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

*CG*

_{2}(

*R*) \

*J*(

*R*) (or

*CG*(

_{J}*R*) for short) where

*CG*

_{2}(

*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.