Matching Points with Squares

  • Authors:

  • Bernardo M. Ábrego;Esther M. Arkin;Silvia Fernández-Merchant;Ferran Hurtado;Mikio Kano;Joseph S. B. Mitchell;Jorge Urrutia

  • Affiliations:

  • California State University, Department of Mathematics, Northridge, CA, USA;State University of New York, Department of Applied Mathematics and Statistics, Stony Brook, NY, USA;California State University, Department of Mathematics, Northridge, CA, USA;Universitat Politècnica de Catalunya, Departament de Matemàtica Aplicada II, Barcelona, Spain;Ibaraki University, Department of Computer and Information Sciences, Ibaraki, Japan;State University of New York, Department of Applied Mathematics and Statistics, Stony Brook, NY, USA;Universidad Nacional Autónoma de México, Instituto de Matemáticas, Mexico City, Mexico

  • Venue:
  • Discrete & Computational Geometry
  • Year:
  • 2009

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Abstract

Given a class $\mathcal{C}$of geometric objects and a point set P, a $\mathcal{C}$-matching of P is a set $M=\{C_{1},\dots,C_{k}\}\subseteq \mathcal{C}$of elements of $\mathcal{C}$such that each C i contains exactly two elements of P and each element of P lies in at most one C i . If all of the elements of P belong to some C i , M is called a perfect matching. If, in addition, all of the elements of M are pairwise disjoint, we say that this matching M is strong. In this paper we study the existence and characteristics of $\mathcal{C}$-matchings for point sets in the plane when $\mathcal{C}$is the set of isothetic squares in the plane. A consequence of our results is a proof that the Delaunay triangulations for the L ∞ metric and the L 1 metric always admit a Hamiltonian path.