# Hanani-Tutte for Radial Planarity II

• Michael Pelsmajer
• Marcus Schaefer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9801)

## 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. A pair of edges e and f in a graph is independent if e and f do not share a vertex.

We show that a graph G is radial planar if G has a radial drawing in which every two independent edges cross an even number of times; the radial embedding has the same leveling as the radial drawing. In other words, we establish the strong Hanani-Tutte theorem for radial planarity. This characterization yields a very simple algorithm for radial planarity testing.

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## Authors and Affiliations

• 1
• Michael Pelsmajer
• 2
• Marcus Schaefer
• 3
Email author
1. 1.IST AustriaKlosterneuburgAustria
2. 2.Illinois Institute of TechnologyChicagoUSA
3. 3.DePaul UniversityChicagoUSA