License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2019.29
URN: urn:nbn:de:0030-drops-104338
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2019/10433/
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Dujmovic, Vida ; Morin, Pat

Dual Circumference and Collinear Sets

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LIPIcs-SoCG-2019-29.pdf (1 MB)

Abstract

We show that, if an n-vertex triangulation T of maximum degree Delta has a dual that contains a cycle of length l, then T has a non-crossing straight-line drawing in which some set, called a collinear set, of Omega(l/Delta^4) vertices lie on a line. Using the current lower bounds on the length of longest cycles in 3-regular 3-connected graphs, this implies that every n-vertex planar graph of maximum degree Delta has a collinear set of size Omega(n^{0.8}/Delta^4). Very recently, Dujmovic et al. (SODA 2019) showed that, if S is a collinear set in a triangulation T then, for any point set X subset R^2 with |X|=|S|, T has a non-crossing straight-line drawing in which the vertices of S are drawn on the points in X. Because of this, collinear sets have numerous applications in graph drawing and related areas.

BibTeX - Entry

@InProceedings{dujmovic_et_al:LIPIcs:2019:10433,
  author =	{Vida Dujmovic and Pat Morin},
  title =	{{Dual Circumference and Collinear Sets}},
  booktitle =	{35th International Symposium on Computational Geometry (SoCG 2019)},
  pages =	{29:1--29:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-104-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{129},
  editor =	{Gill Barequet and Yusu Wang},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2019/10433},
  URN =		{urn:nbn:de:0030-drops-104338},
  doi =		{10.4230/LIPIcs.SoCG.2019.29},
  annote =	{Keywords: Planar graphs, collinear sets, untangling, column planarity, universal point subsets, partial simultaneous geometric drawings}
}

Keywords: Planar graphs, collinear sets, untangling, column planarity, universal point subsets, partial simultaneous geometric drawings
Collection: 35th International Symposium on Computational Geometry (SoCG 2019)
Issue Date: 2019
Date of publication: 11.06.2019

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