Planar, Outerplanar and Ring Graph Cozero-Divisor Graphs

Abstract

Let \(R\) be a commutative ring with non-zero identity. The cozero-divisor graph of \(R\), denoted by \(\Gamma'(R)\), is a graph with vertex-set \(W^*(R)\), which is the set of all non-zero non-unit elements of \(R\), and two distinct vertices \(a\) and \(b\) in \(W^*(R)\) are adjacent if and only if \(a \not\in Rb\) and \(b \not\in Ra\), where for \(c \in R\), \(Rc\) is the ideal generated by \(c\). In this paper, we completely determine all finite commutative rings \(R\) such that \(\Gamma'(R)\) is planar, outerplanar and a ring graph.