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 ${\mathrm{\Gamma }}^{\prime }(R)$, is a graph with vertex-set $W^{\ast }(R)$, which is the set of all non-zero non-unit elements of $R$, and two distinct vertices $a$ and $b$ in $W^{\ast }(R)$ are adjacent if and only if $a\notin Rb$ and $b\notin Ra$, 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 ${\mathrm{\Gamma }}^{\prime }(R)$ is planar, outerplanar and a ring graph.