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Circumscribing Polygons and Polygonizations for Disjoint Line Segments

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

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ACM Subject Classification
  • Theory of computation → Computational geometry
  • Mathematics of computing → Combinatoric problems
Keywords
  • circumscribing polygon
  • Hamiltonicity
  • extremal combinatorics

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Abstract

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.

Cite As Get BibTex

Hugo A. Akitaya, Matias Korman, Mikhail Rudoy, Diane L. Souvaine, and Csaba D. Tóth. Circumscribing Polygons and Polygonizations for Disjoint Line Segments. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 9:1-9:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.SoCG.2019.9

Author Details

Hugo A. Akitaya
  • Department of Computer Science, Tufts University, Medford, MA, USA
Matias Korman
  • Department of Computer Science, Tufts University, Medford, MA, USA
Mikhail Rudoy
  • CSAIL, Massachusetts Institute of Technology, Cambridge, MA, USA
  • Google Inc., Cambridge, MA, USA
Diane L. Souvaine
  • Department of Computer Science, Tufts University, Medford, MA, USA
Csaba D. Tóth
  • Department of Mathematics, California State University Northridge, Los Angeles, CA
  • Department of Computer Science, Tufts University, Medford, MA, USA

Funding

Research supported in part by the NSF awards CCF-1422311 and CCF-1423615.
  • Korman, Matias: Partially supported by MEXT KAKENHI No. 17K12635.
  • Souvaine, Diane L.: Partially supported by the Erwin Schrödinger Institute for Mathematics and Physics (ESI).

References

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