eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2019-06-11
9:1
9:17
10.4230/LIPIcs.SoCG.2019.9
article
Circumscribing Polygons and Polygonizations for Disjoint Line Segments
Akitaya, Hugo A.
1
Korman, Matias
1
Rudoy, Mikhail
2
3
Souvaine, Diane L.
1
Tóth, Csaba D.
4
1
Department of Computer Science, Tufts University, Medford, MA, USA
CSAIL, Massachusetts Institute of Technology, Cambridge, MA, USA
Google Inc., Cambridge, MA, USA
Department of Mathematics, California State University Northridge, Los Angeles, CA
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol129-socg2019/LIPIcs.SoCG.2019.9/LIPIcs.SoCG.2019.9.pdf
circumscribing polygon
Hamiltonicity
extremal combinatorics