Dual Circumference and Collinear Sets

Authors Vida Dujmović, Pat Morin

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Author Details

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


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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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)


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
  • Planar graphs
  • collinear sets
  • untangling
  • column planarity
  • universal point subsets
  • partial simultaneous geometric drawings


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