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2016 | OriginalPaper | Buchkapitel

Hanani-Tutte for Radial Planarity II

verfasst von : Radoslav Fulek, Michael Pelsmajer, Marcus Schaefer

Erschienen in: Graph Drawing and Network Visualization

Verlag: Springer International Publishing

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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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Vorheriges Kapitel A Direct Proof of the Strong Hanani–Tutte Theorem on the Projective Plane

Nächstes Kapitel Beyond Level Planarity

Search for “cylindrical mirror anamorphoses” on the web for many cool pictures of this transformation.

Note that we define an essential curve slightly differently than usual.

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Titel: Hanani-Tutte for Radial Planarity II
verfasst von: Radoslav Fulek
Michael Pelsmajer
Marcus Schaefer
Verlag: Springer International Publishing
Buch: Graph Drawing and Network Visualization
Print ISBN: 978-3-319-50105-5

Electronic ISBN: 978-3-319-50106-2

Copyright-Jahr: 2016
DOI: https://doi.org/10.1007/978-3-319-50106-2_36

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