Hanani-Tutte for Radial Planarity

A drawing of a graph G is radial if the vertices of G are placed on concentric circles C1, . . . , Ck 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. Submitted: November 2015 Reviewed: August 2016 Revised: September 2016 Accepted: November 2016 Final: December 2016 Published: January 2017 Article type: Regular paper Communicated by: E. Di Giacomo and A. Lubiw An earlier version of the paper appeared in the proceedings of Graph Drawing 2015. The research of the first author has received funding from the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) under REA grant agreement no [291734]. E-mail addresses: [email protected] (Radoslav Fulek) [email protected] (Michael Pelsmajer) [email protected] (Marcus Schaefer)