Circumscribing polygons and polygonizations for disjoint line segments

Hugo A. Akitaya, Matias Korman, Mikhail Rudoy, Diane L. Souvaine, Csaba D. Tóth

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

Given a planar straight-line graph G = (V, E) in ℝ2, a circumscribing polygon of G is a simple polygon P whose vertex set is V , and every edge in E is either an edge or an internal diagonal of P. A circumscribing polygon is a polygonization for G if every edge in E is an edge of P. We prove that every arrangement of n disjoint line segments in the plane has a subset of size Ω(√n) that admits a circumscribing polygon, which is the first improvement on this bound in 20 years. We explore relations between circumscribing polygons and other problems in combinatorial geometry, and generalizations to ℝ3. We show that it is NP-complete to decide whether a given graph G admits a circumscribing polygon, even if G is 2-regular. Settling a 30-year old conjecture by Rappaport, we also show that it is NP-complete to determine whether a geometric matching admits a polygonization.

Original languageEnglish
Title of host publication35th International Symposium on Computational Geometry, SoCG 2019
EditorsGill Barequet, Yusu Wang
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959771047
DOIs
Publication statusPublished - 2019 Jun 1
Externally publishedYes
Event35th International Symposium on Computational Geometry, SoCG 2019 - Portland, United States
Duration: 2019 Jun 182019 Jun 21

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume129
ISSN (Print)1868-8969

Conference

Conference35th International Symposium on Computational Geometry, SoCG 2019
CountryUnited States
CityPortland
Period19/6/1819/6/21

Keywords

  • Circumscribing polygon
  • Extremal combinatorics
  • Hamiltonicity

ASJC Scopus subject areas

  • Software

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  • Cite this

    Akitaya, H. A., Korman, M., Rudoy, M., Souvaine, D. L., & Tóth, C. D. (2019). Circumscribing polygons and polygonizations for disjoint line segments. In G. Barequet, & Y. Wang (Eds.), 35th International Symposium on Computational Geometry, SoCG 2019 [9] (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 129). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.SoCG.2019.9