TY - JOUR

T1 - Compatible geometric matchings

AU - Aichholzer, Oswin

AU - Bereg, Sergey

AU - Dumitrescu, Adrian

AU - García, Alfredo

AU - Huemer, Clemens

AU - Hurtado, Ferran

AU - Kano, Mikio

AU - Márquez, Alberto

AU - Rappaport, David

AU - Smorodinsky, Shakhar

AU - Souvaine, Diane

AU - Urrutia, Jorge

AU - Wood, David R.

N1 - Funding Information:
✩ This work was initiated at the 3rd U.P.C. Workshop on Combinatorial Geometry (Caldes de Malavella, Catalunya, Spain, May 2006). Preliminary versions of this paper were presented at the 17th Fall Workshop on Computational and Combinatorial Geometry (FWCG ’07), and at Topological and Geometric Graph Theory (TGGT ’08) published in Electronic Notes in Discrete Mathematics 31 (2008) 201–206. * Corresponding author. E-mail addresses: oaich@ist.TUGraz.at (O. Aichholzer), besp@utdallas.edu (S. Bereg), ad@cs.uwm.edu (A. Dumitrescu), olaverri@unizar.es (A. García), clemens.huemer@upc.edu (C. Huemer), ferran.hurtado@upc.edu (F. Hurtado), kano@mx.ibaraki.ac.jp (M. Kano), almar@us.es (A. Márquez), daver@cs.queensu.ca (D. Rappaport), shakhar@math.bgu.ac.il (S. Smorodinsky), dls@cs.tufts.edu (D. Souvaine), urrutia@math.unam.mx (J. Urrutia), D.Wood@ms.unimelb.edu.au (D.R. Wood). 1 Supported by the Austrian FWF Joint Research Project ‘Industrial Geometry’ S9205-N12. 2 Research partially supported by NSF CAREER Grant CCF-0444188. 3 Research supported by the project MEC MTM2006-01267. 4 Research supported by the projects MEC MTM2006-01267 and DURSI 2005SGR00692. 5 Research supported by NSERC of Canada Discovery Grant 9204. 6 Supported by CONACYT of Mexico, Proyecto SEP-2004-Co1-45876. 7 Supported by a QEII Research Fellowship. Initiated at the Universitat Politècnica de Catalunya, where supported by the Marie Curie Fellowship MEIF-CT-2006-023865.

PY - 2009/8/1

Y1 - 2009/8/1

N2 - This paper studies non-crossing geometric perfect matchings. Two such perfect matchings are compatible if they have the same vertex set and their union is also non-crossing. Our first result states that for any two perfect matchings M and M′ of the same set of n points, for some k∈O(logn), there is a sequence of perfect matchings M= M0, M1,⋯, Mk= M′, such that each Mi is compatible with Mi+ 1. This improves the previous best bound of k≤n-2. We then study the conjecture: every perfect matching with an even number of edges has an edge-disjoint compatible perfect matching. We introduce a sequence of stronger conjectures that imply this conjecture, and prove the strongest of these conjectures in the case of perfect matchings that consist of vertical and horizontal segments. Finally, we prove that every perfect matching with n edges has an edge-disjoint compatible matching with approximately 4n/5 edges.

AB - This paper studies non-crossing geometric perfect matchings. Two such perfect matchings are compatible if they have the same vertex set and their union is also non-crossing. Our first result states that for any two perfect matchings M and M′ of the same set of n points, for some k∈O(logn), there is a sequence of perfect matchings M= M0, M1,⋯, Mk= M′, such that each Mi is compatible with Mi+ 1. This improves the previous best bound of k≤n-2. We then study the conjecture: every perfect matching with an even number of edges has an edge-disjoint compatible perfect matching. We introduce a sequence of stronger conjectures that imply this conjecture, and prove the strongest of these conjectures in the case of perfect matchings that consist of vertical and horizontal segments. Finally, we prove that every perfect matching with n edges has an edge-disjoint compatible matching with approximately 4n/5 edges.

KW - Compatible matching

KW - Convex-hull-connected segments

KW - Convexly independent segments

KW - Geometric graph

KW - Segments in convex position

UR - http://www.scopus.com/inward/record.url?scp=70849104788&partnerID=8YFLogxK

U2 - 10.1016/j.comgeo.2008.12.005

DO - 10.1016/j.comgeo.2008.12.005

M3 - Article

AN - SCOPUS:70849104788

VL - 42

SP - 617

EP - 626

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

SN - 0925-7721

IS - 6-7

ER -