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

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LIPIcs.SoCG.2022.5.pdf
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## Cite As

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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