Arrangements of pseudocircles: Triangles and drawings

Stefan Felsner, Manfred Scheucher

Research output: Chapter in Book/Report/Conference proceedingConference contributionResearchpeer-review

Abstract

A pseudocircle is a simple closed curve on the sphere or in the plane. The study of arrangements of pseudocircles was initiated by Grünbaum, who defined them as collections of simple closed curves that pairwise intersect in exactly two crossings. Grünbaum conjectured that the number of triangular cells p3 in digon-free arrangements of n pairwise intersecting pseudocircles is at least 2n - 4 We present examples to disprove this conjecture. With a recursive construction based on an example with 12 pseudocircles and 16 triangles we obtain a family with p3(A)/n → 16/11 = 1.45. We expect that the lower bound p3(A) ≥ 4n/3 is tight for infinitely many simple arrangements. It may however be that digon-free arrangements of n pairwise intersecting circles indeed have at least 2n - 4 triangles. For pairwise intersecting arrangements with digons we have a lower bound of p3 ≥ 2n/3, and conjecture that p3 ≥ n - 1. Concerning the maximum number of triangles in pairwise intersecting arrangements of pseudocircles, we show that p3 ≤ 2n2/3+O(n). This is essentially best possible because families of pairwise intersecting arrangements of n pseudocircles with p3/n2 → 2/3 as n→∞ are known. The paper contains many drawings of arrangements of pseudocircles and a good fraction of these drawings was produced automatically from the combinatorial data produced by the generation algorithm. In the final section we describe some aspects of the drawing algorithm.

LanguageEnglish
Title of host publicationGraph Drawing and Network Visualization - 25th International Symposium, GD 2017, Revised Selected Papers
PublisherSpringer Verlag Heidelberg
Pages127-139
Number of pages13
ISBN (Print)9783319739144
DOIs
StatusPublished - 1 Jan 2018
Event25th International Symposium on Graph Drawing and Network Visualization, GD 2017 - Boston, United States
Duration: 25 Sep 201727 Sep 2017

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume10692 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference25th International Symposium on Graph Drawing and Network Visualization, GD 2017
CountryUnited States
CityBoston
Period25/09/1727/09/17

Fingerprint

Triangle
Arrangement
Pairwise
Simple Closed Curve
Lower bound
Disprove
Drawing
Intersect
Triangular
Circle
Cell

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

Cite this

Felsner, S., & Scheucher, M. (2018). Arrangements of pseudocircles: Triangles and drawings. In Graph Drawing and Network Visualization - 25th International Symposium, GD 2017, Revised Selected Papers (pp. 127-139). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 10692 LNCS). Springer Verlag Heidelberg. https://doi.org/10.1007/978-3-319-73915-1_11

Arrangements of pseudocircles : Triangles and drawings. / Felsner, Stefan; Scheucher, Manfred.

Graph Drawing and Network Visualization - 25th International Symposium, GD 2017, Revised Selected Papers. Springer Verlag Heidelberg, 2018. p. 127-139 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 10692 LNCS).

Research output: Chapter in Book/Report/Conference proceedingConference contributionResearchpeer-review

Felsner, S & Scheucher, M 2018, Arrangements of pseudocircles: Triangles and drawings. in Graph Drawing and Network Visualization - 25th International Symposium, GD 2017, Revised Selected Papers. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 10692 LNCS, Springer Verlag Heidelberg, pp. 127-139, 25th International Symposium on Graph Drawing and Network Visualization, GD 2017, Boston, United States, 25/09/17. https://doi.org/10.1007/978-3-319-73915-1_11
Felsner S, Scheucher M. Arrangements of pseudocircles: Triangles and drawings. In Graph Drawing and Network Visualization - 25th International Symposium, GD 2017, Revised Selected Papers. Springer Verlag Heidelberg. 2018. p. 127-139. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). https://doi.org/10.1007/978-3-319-73915-1_11
Felsner, Stefan ; Scheucher, Manfred. / Arrangements of pseudocircles : Triangles and drawings. Graph Drawing and Network Visualization - 25th International Symposium, GD 2017, Revised Selected Papers. Springer Verlag Heidelberg, 2018. pp. 127-139 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
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