Graphs and Combinatorics

, Volume 23, Supplement 1, pp 85–98 | Cite as

Bichromatic Quadrangulations with Steiner Points

  • Victor Alvarez
  • Toshinori Sakai
  • Jorge Urrutia


Let P be a k colored point set in general position, k  ≥  2. A family of quadrilaterals with disjoint interiors \(\mathcal Q_1, \ldots, \mathcal Q_m\) is called a quadrangulation of P if \(V(\mathcal Q_1) \cup \cdots \cup V(\mathcal Q_m) = P\) , the edges of all \(\mathcal Q_i\) join points with different colors, and \(\mathcal Q_1 \cup \cdots \cup \mathcal Q_m = {\rm Conv} (P)\) . In general it is easy to see that not all k-colored point sets admit a quadrangulation; when they do, we call them quadrangulatable. For a point set to be quadrangulatable it must satisfy that its convex hull Conv(P) has an even number of points and that consecutive vertices of Conv(P) receive different colors. This will be assumed from now on. In this paper, we study the following type of questions: Let P be a k-colored point set. How many Steiner points in the interior of Conv(P) do we need to add to P to make it quadrangulatable? When k = 2, we usually call P a bichromatic point set, and its color classes are usually denoted by R and B, i.e. the red and blue elements of P. In this paper, we prove that any bichromatic point set \(P = R\cup B\) where |R|  = |B|  =  n can be made quadrangulatable by adding at most \(\left\lfloor\frac{n -1}{3}\right\rfloor + \left\lfloor\frac{n}{2}\right\rfloor + 1\) Steiner points and that \(\frac{m}{3}\) Steiner points are occasionally necessary. To prove the latter, we also show that the convex hull of any monochromatic point set P of n elements can be always partitioned into a set \({\mathcal{S}} = \{{\mathcal{S}}_1,\ldots, {\mathcal{S}}_{t}\}\) of star-shaped polygons with disjoint interiors, where \(V({\mathcal{S}}_1)\cup\cdots\cup V({\mathcal{S}}_{t }) = P\) , and \(t\leq \left\lfloor\frac{n - 1}{3}\right\rfloor+1\) . For n  =  3k this bound is tight. Finally, we prove that there are 3-colored point sets that cannot be completed to 3-quadrangulatable point sets.


Triangulations Quadrangulations Bicolored point sets Steiner points 


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  1. Bondy, A., Murty, U. S. R.: Graph Theory with Applications, Elsevier Science Publishing, New York (1976)Google Scholar
  2. Bose, P., Toussaint, G.: Characterizing and efficiently computing quadrangulations of planar point sets. Comput. Aided Geom. Des. 14, 763–785 (1997)Google Scholar
  3. Bremner, D., Hurtado, F., Ramaswami, S., Sacristán, V.: Small convex quadrangulations of point sets. Algorithmica 38(2), 317–339 (2003)Google Scholar
  4. Cortés, C., Márquez, A., Nakamoto, A., Valenzula, J.: Quadrangulations and 2-colorations. 21st European Workshop on Computational Geometry, pp. 65–68. Eindhoven, March 9–11, 2005Google Scholar
  5. Fisk, S.: A short proof of Chvátal’s watchman theorem. J. Comb. Theory B 24, 374 (1978)Google Scholar
  6. Heredia, M., Urrutia, J.: On convex quadrangulations of point sets on the plane. Discrete geometry, Combinatorics and Graph Theory, Lecture Notes in Computer Science 4381, 38–46 (2007)Google Scholar
  7. Lai, M. L., Schumaker, L. L.: Scattered data interpolation using piecewise polynomials of degree six. SIAM J. Numer. Anal. 34, 905–921 (1997)Google Scholar
  8. Ramaswami, S., Ramos, P., Toussaint, G.: Converting triangulations to quadrangulations. Comput. Geom. Theory Appl. 9, 257–276 (1998)Google Scholar
  9. Sperner, E.: Neuer Beweis für die Invarianz der Dimensionszahl und des Gebietes. Hambg. Abhand. 6, 265–272 (1928).Google Scholar
  10. Toussaint, G.: Quadrangulations of planar sets. In: Proceedings of the 4th International Workshop on Algorithms and Data Structures, pp. 218–227. Springer, Berlin (1995)Google Scholar

Copyright information

© Springer-Verlag Tokyo 2007

Authors and Affiliations

  • Victor Alvarez
    • 1
  • Toshinori Sakai
    • 2
  • Jorge Urrutia
    • 3
  1. 1.Facultad de CienciasUniversidad Nacional Autónoma de MéxicoMéxico D.FMéxico
  2. 2.Research Institute of Educational DevelopmentTokai UniversityTokyoJapan
  3. 3.Instituto de MatemáticasUniversidad Nacional Autónoma de MéxicoMéxico D.FMéxico

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