License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2023.17
URN: urn:nbn:de:0030-drops-178676
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2023/17867/
Go to the corresponding LIPIcs Volume Portal

Bergold, Helena ; Felsner, Stefan ; Scheucher, Manfred

An Extension Theorem for Signotopes

pdf-format:
LIPIcs-SoCG-2023-17.pdf (0.8 MB)

Abstract

In 1926, Levi showed that, for every pseudoline arrangement ? and two points in the plane, ? can be extended by a pseudoline which contains the two prescribed points. Later extendability was studied for arrangements of pseudohyperplanes in higher dimensions. While the extendability of an arrangement of proper hyperplanes in ℝ^d with a hyperplane containing d prescribed points is trivial, Richter-Gebert found an arrangement of pseudoplanes in ℝ³ which cannot be extended with a pseudoplane containing two particular prescribed points.
In this article, we investigate the extendability of signotopes, which are a combinatorial structure encoding a rich subclass of pseudohyperplane arrangements. Our main result is that signotopes of odd rank are extendable in the sense that for two prescribed crossing points we can add an element containing them. Moreover, we conjecture that in all even ranks r ≥ 4 there exist signotopes which are not extendable for two prescribed points. Our conjecture is supported by examples in ranks 4, 6, 8, 10, and 12 that were found with a SAT based approach.

BibTeX - Entry

@InProceedings{bergold_et_al:LIPIcs.SoCG.2023.17,
  author =	{Bergold, Helena and Felsner, Stefan and Scheucher, Manfred},
  title =	{{An Extension Theorem for Signotopes}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{17:1--17:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2023/17867},
  URN =		{urn:nbn:de:0030-drops-178676},
  doi =		{10.4230/LIPIcs.SoCG.2023.17},
  annote =	{Keywords: arrangement of pseudolines, extendability, Levi’s extension lemma, arrangement of pseudohyperplanes, signotope, oriented matroid, partial order, Boolean satisfiability (SAT)}
}

Keywords: arrangement of pseudolines, extendability, Levi’s extension lemma, arrangement of pseudohyperplanes, signotope, oriented matroid, partial order, Boolean satisfiability (SAT)
Collection: 39th International Symposium on Computational Geometry (SoCG 2023)
Issue Date: 2023
Date of publication: 09.06.2023
Supplementary Material: Software (Data and Source Code): https://github.com/manfredscheucher/supplemental-signotope-extension

DROPS-Home | Fulltext Search | Imprint | Privacy Published by LZI