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# Dual Circumference and Collinear Sets

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LIPIcs.SoCG.2019.29.pdf
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## Acknowledgements

Much of this research took place during the Sixth Workshop on Order and Geometry held in Ciążeń, Poland, September 19 - 22, 2018. The authors are grateful to the organizers, Stefan Felsner and Piotr Micek, and to the other participants for providing a stimulating research environment.

## Cite As

Vida Dujmović and Pat Morin. Dual Circumference and Collinear Sets. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 29:1-29:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.SoCG.2019.29

## 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, Dujmović 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.

## Subject Classification

##### ACM Subject Classification
• Mathematics of computing → Graph theory
• Mathematics of computing → Extremal graph theory
• Human-centered computing → Graph drawings
##### Keywords
• Planar graphs
• collinear sets
• untangling
• column planarity
• universal point subsets
• partial simultaneous geometric drawings

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