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

Authors Oswin Aichholzer , Alfredo García , Javier Tejel , Birgit Vogtenhuber , Alexandra Weinberger



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Author Details

Oswin Aichholzer
  • Institute of Software Technology, Technische Universität Graz, Austria
Alfredo García
  • Departamento de Métodos Estadísticos and IUMA, University of Zaragoza, Spain
Javier Tejel
  • Departamento de Métodos Estadísticos and IUMA, University of Zaragoza, Spain
Birgit Vogtenhuber
  • Institute of Software Technology, Technische Universität Graz, Austria
Alexandra Weinberger
  • Institute of Software Technology, Technische Universität Graz, Austria

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Oswin Aichholzer, Alfredo García, Javier Tejel, Birgit Vogtenhuber, and Alexandra Weinberger. Twisted Ways to Find Plane Structures in Simple Drawings of Complete Graphs. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 5:1-5:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.SoCG.2022.5

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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorics
  • Mathematics of computing → Graph theory
Keywords
  • Simple drawings
  • simple topological graphs
  • disjoint edges
  • plane matching
  • plane path

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References

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