Colored Quadrangulations with Steiner Points

  • Victor Alvarez
  • Atsuhiro Nakamoto
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8296)


Let P ⊂ ℝ2 be a k-colored set of n points in general position, where k ≥ 2. A k-colored quadrangulation on P is a properly colored straight-edge plane graph G with vertex set P such that the boundary of the unbounded face of G coincides with CH(P) and that each bounded face of G is quadrilateral, where CH(P) stands for the boundary of the convex hull of P. It is easily checked that in general not every k-colored P admits a k-colored quadrangulation, and hence we need the use of Steiner points, that is, auxiliary points which can be put in any position of the interior of the convex hull of P and can have any color among the k colors. In this paper, we show that if P satisfies some condition for colors of the points in the convex hull, then a k-colored quadrangulation of P can always be constructed using less than \( \frac{(16 k-2) n+7 k-2}{39 k-6}\) Steiner points. Our upper bound improves the known upper bound for k = 3, and gives the first bounds for k ≥ 4.


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  1. 1.
    Alvarez, V., Sakai, T., Urrutia, J.: Bichromatic quadrangulations with steiner points. Graph. Combin. 23, 85–98 (2007)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bremner, D., Hurtado, F., Ramaswami, S., Sacristan, V.: Small strictly convex quadrilateral meshes of point sets. Algorithmica 38, 317–339 (2003)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bose, P., Toussaint, G.T.: Characterizing and efficiently computing quadrangulations of planar point sets. Computer Aided Geometric Design 14, 763–785 (1997)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bose, P., Ramaswami, S., Toussaint, G.T., Turki, A.: Experimental results on quadrangulations of sets of fixed points. Computer Aided Geometric Design 19, 533–552 (2007)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Cortés, C., Márquez, A., Nakamoto, A., Valenzuela, J.: Quadrangulations and 2-colorations. In: EuroCG, pp. 65–68. Technische Universiteit Eindhoven (2005)Google Scholar
  6. 6.
    Heredia, V.M., Urrutia, J.: On convex quadrangulations of point sets on the plane. In: Akiyama, J., Chen, W.Y.C., Kano, M., Li, X., Yu, Q. (eds.) CJCDGCGT 2005. LNCS, vol. 4381, pp. 38–46. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  7. 7.
    Kato, S., Mori, R., Nakamoto, A.: Quadrangulations on 3-colored point sets with steiner points and their winding number. Graphs Combin., DOI 10.1007/s00373-013-1346-4Google Scholar
  8. 8.
    Lai, M.-J., Schumaker, L.L.: Scattered data interpolation using c2 supersplines of degree six. SIAM J. Numer. Anal. 34, 905–921 (1997)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Ramaswami, S., Ramos, P.A., Toussaint, G.T.: Converting triangulations to quadrangulations. Comput. Geom. 9, 257–276 (1998)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Schiffer, T., Aurenhammer, F., Demuth, M.: Computing convex quadrangulations. Discrete Applied Math. 160, 648–656 (2012)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Toussaint, G.T.: Quadrangulations of planar sets. In: Sack, J.-R., Akl, S.G., Dehne, F., Santoro, N. (eds.) WADS 1995. LNCS, vol. 955, pp. 218–227. Springer, Heidelberg (1995)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Victor Alvarez
    • 1
  • Atsuhiro Nakamoto
    • 2
  1. 1.Fachrichtung InformatikUniversität des SaarlandesSaarbrückenGermany
  2. 2.Department of MathematicsYokohama National UniversityYokohamaJapan

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