Compatible geometric matchings

Oswin Aichholzer, Sergey Bereg, Adrian Dumitrescu, Alfredo García, Clemens Huemer, Ferran Hurtado, Mikio Kano, Alberto Márquez, David Rappaport, Shakhar Smorodinsky, Diane Souvaine, Jorge Urrutia, David R. Wood

Research output: Contribution to journalArticlepeer-review

31 Scopus citations

Abstract

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.

Original languageEnglish
Pages (from-to)617-626
Number of pages10
JournalComputational Geometry: Theory and Applications
Volume42
Issue number6-7
DOIs
StatePublished - 1 Aug 2009

Keywords

  • Compatible matching
  • Convex-hull-connected segments
  • Convexly independent segments
  • Geometric graph
  • Segments in convex position

ASJC Scopus subject areas

  • Computer Science Applications
  • Geometry and Topology
  • Control and Optimization
  • Computational Theory and Mathematics
  • Computational Mathematics

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