License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2019.9
URN: urn:nbn:de:0030-drops-104136
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Akitaya, Hugo A. ; Korman, Matias ; Rudoy, Mikhail ; Souvaine, Diane L. ; Tóth, Csaba D.

Circumscribing Polygons and Polygonizations for Disjoint Line Segments

LIPIcs-SoCG-2019-9.pdf (0.9 MB)


Given a planar straight-line graph G=(V,E) in R^2, 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 Omega(sqrt{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 R^3.
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.

BibTeX - Entry

  author =	{Hugo A. Akitaya and Matias Korman and Mikhail Rudoy and Diane L. Souvaine and Csaba D. T{\'o}th},
  title =	{{Circumscribing Polygons and Polygonizations for Disjoint Line Segments}},
  booktitle =	{35th International Symposium on Computational Geometry (SoCG 2019)},
  pages =	{9:1--9:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-104-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{129},
  editor =	{Gill Barequet and Yusu Wang},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-104136},
  doi =		{10.4230/LIPIcs.SoCG.2019.9},
  annote =	{Keywords: circumscribing polygon, Hamiltonicity, extremal combinatorics}

Keywords: circumscribing polygon, Hamiltonicity, extremal combinatorics
Collection: 35th International Symposium on Computational Geometry (SoCG 2019)
Issue Date: 2019
Date of publication: 11.06.2019

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