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The Number of Triangulations on Planar Point Sets

  • Emo Welzl
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4372)

Abstract

We give a brief account of results concerning the number of triangulations on finite point sets in the plane, both for arbitrary sets and for specific sets such as the n ×n integer lattice.

Keywords

Random Walk Polynomial Time Computational Geometry Geometric Graph Integer Lattice 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Aichholzer, O.: The path of a triangulation. In: Proc.15th Ann. ACM Symp. on Computational Geometry, pp. 14–23. ACM Press, New York (1999)Google Scholar
  2. 2.
    Aichholzer, O., Hackl, T., Krasser, H., Huemer, C., Hurtado, F., Vogtenhuber, B.: On the number of plane graphs. In: Proc. 17th Ann. ACM-SIAM Symp. on Discrete Algorithms, pp. 504–513. ACM Press, New York (2006)CrossRefGoogle Scholar
  3. 3.
    Ajtai, M., Chvátal, V., Newborn, M.M., Szemerédi, E.: Crossing-free subgraphs. Annals Discrete Math. 12, 9–12 (1982)MATHGoogle Scholar
  4. 4.
    Anclin, E.E.: An upper bound for the number of planar lattice triangulations. J.Combinat.Theory,Ser.A 103, 383–386 (2003)CrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    Avisand, D., Fukuda, K.: Reverse search for enumeration. Discrete Appl. Math. 65, 21–46 (1996)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Dennyand, M.O., Sohler, C.A.: Encoding a triangulation as a permutation of its point set. In: Proc. 9th Canadian Conf. on Computational Geometry, pp. 39–43 (1997)Google Scholar
  7. 7.
    García, A., Noy, M., Tejel, J.: Lower bounds on the number of crossing-free subgraphs of K N. Comput.Geom.TheoryAppl. 16, 211–221 (2000)CrossRefMATHGoogle Scholar
  8. 8.
    Hurtadoand, F., Noy, M.: Counting triangulations of almost-convex polygons. Ars Combinatorica 45, 169–179 (1997)Google Scholar
  9. 9.
    Kaibeland, V., Ziegler, G.: Counting lattice triangulations. In: Wensley, C.D. (ed.) British Combinatorial Surveys, Cambridge University Press, Cambridge (2003)Google Scholar
  10. 10.
    Matoušek, J., Valtr, P., Welzl, E.: On two encodings of lattice triangulations. Manuscript (2006)Google Scholar
  11. 11.
    McShineand, L., Tetali, P.: On the mixing time of the triangulation walk and other Catalan structures. In: Pardalos, P.M., Rajasekaran, S., Rolim, J. (eds.) DIMACS-AMS volume on Randomization Methods in Algorithm Design. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 43, pp. 147–160 (1998)Google Scholar
  12. 12.
    Molloy, M., Reed, B., Steiger, W.: On the mixing rate of the triangulation walk. In: Pardalos, P.M., Rajasekaran, S., Rolim, J. (eds.) DIMACS-AMS volume on Randomization Methods in Algorithm Design. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 43, pp. 179–190 (1998)Google Scholar
  13. 13.
    Orevkov, S.Y.: A symptotic number of triangulations in Z2. J. Combinat. Theory Ser. A 86, 200–203 (1999)CrossRefMathSciNetMATHGoogle Scholar
  14. 14.
    Pólya, G.: On picture-writing. The American Mathematical Monthly 63, 689–697 (1956), Also in: Alexanderson, G.L.: The Random Walks of George Pólya, MAA, 2000CrossRefMATHGoogle Scholar
  15. 15.
    Sharir, M., Welzl, E.: Random triangulations of planar point sets. In: Proc. 22nd Ann. ACM Symp. on Computational Geometry, pp. 273–281. ACM, New York (2006)Google Scholar
  16. 16.
    Santos, F., Seidel, R.: A better upper bound on the number of triangulations of a planar point set. J. Combinat. Theory Ser. A 102, 186–193 (2003)CrossRefMathSciNetMATHGoogle Scholar
  17. 17.
    Seidel, R.: On the number of triangulations of planar point sets. Combinatorica 18, 297–299 (1998)CrossRefMathSciNetMATHGoogle Scholar
  18. 18.
    Smith, W.S.: Studies in Computational Geometry Motivated by Mesh Generation. Ph.D. Thesis, Princeton University (1989)Google Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Emo Welzl
    • 1
  1. 1.Institute of Theoretical Computer Science, ETH ZurichSwitzerland

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