Compatible geometric matchings

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(log n), 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 3n/4 edges. Institute for Software Technology, Graz University of Technology, Austria ([email protected]) Department of Computer Science, University of Texas at Dallas, U.S.A. ([email protected]) Department of Computer Science, University of Wisconsin-Milwaukee, U.S.A. ([email protected]). Research partially supported by NSF CAREER grant CCF-0444188. Departamento de Métodos Estad́ısticos, Universidad de Zaragoza, Spain ([email protected]) Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Spain ({ferran.hurtado,clemens.huemer,david.wood}@upc.edu). Research supported by the projects MEC MTM2006-01267 and DURSI 2005SGR00692. The research of David Wood is supported by a Marie Curie Fellowship of the European Commission under contract MEIF-CT-2006-023865. Department of Computer and Information Sciences, Ibaraki University, Japan ([email protected]) Departamento de Matemática Aplicada I, Universidad de Sevilla, Spain ([email protected]) Institute of Mathematics, Hebrew University, Israel ([email protected]) Department of Computer Science, Tufts University, U.S.A. ([email protected]) Instituto de Matemáticas, Universidad Nacional Autónoma de México, México ([email protected]). Supported by CONACYT of Mexico, Proyecto SEP-2004-Co1-45876.