TY - GEN

T1 - Circumscribing polygons and polygonizations for disjoint line segments

AU - Akitaya, Hugo A.

AU - Korman, Matias

AU - Rudoy, Mikhail

AU - Souvaine, Diane L.

AU - Tóth, Csaba D.

N1 - Funding Information:
Funding Research supported in part by the NSF awards CCF-1422311 and CCF-1423615. Matias Korman: Partially supported by MEXT KAKENHI No. 17K12635. Diane L. Souvaine: Partially supported by the Erwin Schrödinger Institute for Mathematics and Physics (ESI).
Publisher Copyright:
© H. A. Akitaya, M. Korman, M. Rudoy, C. D. Tóth, and D. L. Souvaine.

PY - 2019/6/1

Y1 - 2019/6/1

N2 - Given a planar straight-line graph G = (V, E) in ℝ2, 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 ℝ3. 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.

AB - Given a planar straight-line graph G = (V, E) in ℝ2, 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 ℝ3. 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.

KW - Circumscribing polygon

KW - Extremal combinatorics

KW - Hamiltonicity

UR - http://www.scopus.com/inward/record.url?scp=85068054873&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85068054873&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.SoCG.2019.9

DO - 10.4230/LIPIcs.SoCG.2019.9

M3 - Conference contribution

AN - SCOPUS:85068054873

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 35th International Symposium on Computational Geometry, SoCG 2019

A2 - Barequet, Gill

A2 - Wang, Yusu

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 35th International Symposium on Computational Geometry, SoCG 2019

Y2 - 18 June 2019 through 21 June 2019

ER -