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Dual Circumference and Collinear Sets

Authors Vida Dujmović, Pat Morin

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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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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.

Cite As Get BibTex

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

Author Details

Vida Dujmović
  • School of Computer Science and Electrical Engineering, University of Ottawa, Canada
Pat Morin
  • School of Computer Science, Carleton University, Canada

Funding

This work was partly funded by NSERC and MRI.

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.

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