License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2022.5
URN: urn:nbn:de:0030-drops-160136
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/16013/
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Aichholzer, Oswin ; García, Alfredo ; Tejel, Javier ; Vogtenhuber, Birgit ; Weinberger, Alexandra

Twisted Ways to Find Plane Structures in Simple Drawings of Complete Graphs

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LIPIcs-SoCG-2022-5.pdf (2 MB)

Abstract

Simple drawings are drawings of graphs in which the edges are Jordan arcs and each pair of edges share at most one point (a proper crossing or a common endpoint). We introduce a special kind of simple drawings that we call generalized twisted drawings. A simple drawing is generalized twisted if there is a point O such that every ray emanating from O crosses every edge of the drawing at most once and there is a ray emanating from O which crosses every edge exactly once.
Via this new class of simple drawings, we show that every simple drawing of the complete graph with n vertices contains Ω(n^{1/2}) pairwise disjoint edges and a plane path of length Ω((log n)/(log log n)). Both results improve over previously known best lower bounds. On the way we show several structural results about and properties of generalized twisted drawings. We further present different characterizations of generalized twisted drawings, which might be of independent interest.

BibTeX - Entry

@InProceedings{aichholzer_et_al:LIPIcs.SoCG.2022.5,
  author =	{Aichholzer, Oswin and Garc{\'\i}a, Alfredo and Tejel, Javier and Vogtenhuber, Birgit and Weinberger, Alexandra},
  title =	{{Twisted Ways to Find Plane Structures in Simple Drawings of Complete Graphs}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{5:1--5:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-227-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{224},
  editor =	{Goaoc, Xavier and Kerber, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/16013},
  URN =		{urn:nbn:de:0030-drops-160136},
  doi =		{10.4230/LIPIcs.SoCG.2022.5},
  annote =	{Keywords: Simple drawings, simple topological graphs, disjoint edges, plane matching, plane path}
}

Keywords: Simple drawings, simple topological graphs, disjoint edges, plane matching, plane path
Collection: 38th International Symposium on Computational Geometry (SoCG 2022)
Issue Date: 2022
Date of publication: 01.06.2022

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