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On Edge-Disjoint Empty Triangles of Point Sets

In: Thirty Essays on Geometric Graph Theory

Author

Listed:
  • Javier Cano

    (Universidad Nacional Autónoma de México, Posgrado en Ciencia e Ingeniería de la Computación)

  • Luis F. Barba

    (Universidad Nacional Autónoma de México, Posgrado en Ciencia e Ingeniería de la Computación)

  • Toshinori Sakai

    (Tokai University, Research Institute of Educational Development)

  • Jorge Urrutia

    (Universidad Nacional Autónoma de México, Instituto de Matemáticas
    Partially supported by CONACYT of México grant number: CB-2012-01-0178379)

Abstract

Let P be a set of points in the plane in general position. Any three points $$x,y,z \in P$$ determine a triangle $$\Delta (x,y,z)$$ of the plane. We say that $$\Delta (x,y,z)$$ is empty if its interior contains no element of P. In this chapter, we study the following problems: What is the size of the largest family of edge-disjoint triangles of a point set? How many triangulations of P are needed to cover all the empty triangles of P? We also study the following problem: What is the largest number of edge-disjoint triangles of P containing a point q of the plane in their interior? We establish upper and lower bounds for these problems.

Suggested Citation

  • Javier Cano & Luis F. Barba & Toshinori Sakai & Jorge Urrutia, 2013. "On Edge-Disjoint Empty Triangles of Point Sets," Springer Books, in: János Pach (ed.), Thirty Essays on Geometric Graph Theory, pages 83-100, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4614-0110-0_7
    DOI: 10.1007/978-1-4614-0110-0_7
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