Hanani-Tutte for Radial Planarity

Abstract

A drawing of a graph $G$ is radial if the vertices of $G$ are placed on concentric circles $C_1, \ldots, C_k$ with common center $c$, and edges are drawn radially: every edge intersects every circle centered at $c$ at most once. $G$ is radial planar if it has a radial embedding, that is, a crossing-free radial drawing. If the vertices of $G$ are ordered or partitioned into ordered levels (as they are for leveled graphs), we require that the assignment of vertices to circles corresponds to the given ordering or leveling. We show that a graph $G$ is radial planar if $G$ has a radial drawing in which every two edges cross an even number of times; the radial embedding has the same leveling as the radial drawing. In other words, we establish the weak variant of the Hanani-Tutte theorem for radial planarity. This generalizes a result by Pach and Tóth.