Planar compressed zero-divisor graphs

Abstract

Let [Formula: see text] be a commutative ring. The relation on [Formula: see text] given by [Formula: see text] if and only if [Formula: see text] is an equivalence relation. The compressed zero-divisor graph, denoted by [Formula: see text], is the graph whose vertices are the equivalence classes induced by [Formula: see text] other than [Formula: see text] and [Formula: see text], such that two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. In this paper, we investigate when [Formula: see text] is planar. First, we characterize all finite non-local rings whose compressed zero-divisor graphs are planar. In the local case, we characterize all graphs that are realized as the planar compressed zero-divisor graphs in some cases. Finally, we find all graphs with at most five vertices that are realized as the compressed zero-divisor graphs.