### 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 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 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.

Original language | English |
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Title of host publication | 35th International Symposium on Computational Geometry, SoCG 2019 |

Editors | Gill Barequet, Yusu Wang |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959771047 |

DOIs | |

Publication status | Published - 2019 Jun 1 |

Externally published | Yes |

Event | 35th International Symposium on Computational Geometry, SoCG 2019 - Portland, United States Duration: 2019 Jun 18 → 2019 Jun 21 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 129 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 35th International Symposium on Computational Geometry, SoCG 2019 |
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Country | United States |

City | Portland |

Period | 19/6/18 → 19/6/21 |

### Keywords

- Circumscribing polygon
- Extremal combinatorics
- Hamiltonicity

### ASJC Scopus subject areas

- Software

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## Cite this

*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