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# Circumscribing Polygons and Polygonizations for Disjoint Line Segments

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## Cite As

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

## 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.

## Subject Classification

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

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## References

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