2016 | OriginalPaper | Chapter
Hanani-Tutte for Radial Planarity II
Authors: Radoslav Fulek, Michael Pelsmajer, Marcus Schaefer
Publisher: Springer International Publishing
Published in:
Graph Drawing and Network Visualization
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.
