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.
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 -