» On planarity of compressed zero-divisor graphs associated to commutative rings

The equivalence class $[r]$ of an element $r\in \mathsf{R}$ is the set of zero-divisors $s$ such that $ann(r)=ann(s)$ , that is, $[r]= \{s\in \mathsf{R} : ann(r)=ann(s)$ . The compressed zero-divisor graph, denoted by $\Gamma_c(\mathsf{R})$ , is the compression of a zero-divisor graph, in which the vertex set is the set of all equivalence classes of nonzero zero-divisors of a ring $\mathsf{R}$ , that is, the vertex set of $\Gamma_c(\mathsf{R})$ is $\mathsf{R}_e -\{[0], [1]\}$ , where $\mathsf{R}_e =\{[r] : r\in \mathsf{R}\}$ and two distinct equivalence classes $[r]$ and $[s]$ are adjacent if and only if $rs = 0$ . In this article, we investigate the planarity of $\Gamma_c(\mathsf{R})$ for some finite local rings of order $p^2$ , $p^3$ and determine the planarity of compressed zero-divisor graph of some local rings of order $32$ , whose zero-divisor graph is nonplanar. Further, we determine values of $m$ and $n$ for which $\Gamma_c(\mathbb{Z}_n)$ and $\Gamma_c(\mathbb{Z}_n[x]/(x^m))$ are planar.