Towards Compatible Triangulations

  • Oswin Aichholzer
  • Franz Aurenhammer
  • Hannes Krasser
  • Ferran Hurtado
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2108)


We state the following conjecture: any two planar n-point sets (that agree on the number of convex hull points) can be triangulated in a compatible manner, i.e., such that the resulting two planar graphs are isomorphic. The conjecture is proved true for point sets with at most three interior points. We further exhibit a class of point sets which can be triangulated compatibly with any other set (that satis?es the obvious size and hull restrictions). Finally, we prove that adding a small number of Steiner points (the number of interior points minus two) always allows for compatible triangulations.


Extreme Point Interior Point Convex Polygon Order Type Steiner Point 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    O. Aichholzer, F. Aurenhammer, H. Krasser, Enumerating order types for small point sets with applications. 17th Ann. ACM Symp. Computational Geometry, Medford, MA, 2001 (to be presented).Google Scholar
  2. 2.
    B. Aronov, R. Seidel, D. Souvaine, On compatible triangulations of simple polygons. Computational Geometry: Theory and Applications 3 (1993), 27–35.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    M. Bern, A. Sahai, Isomorphic triangulations and map conflaation. Personal communication, 2000.Google Scholar
  4. 4.
    J.E. Goodman, R. Pollack, Multidimensional sorting. SIAM J. Computing 12 (1983), 484–507.zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    E. Kranakis, J. Urrutia, Isomorphic triangulations with small number of Steiner points. Int’l J. Computaional Geometry & Applications 9 (1999), 171–180.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    H. Krasser, Kompatible Triangulierungen ebener Punktmengen. Master Thesis, Institute for Theoretical Computer Science, Graz University of Technology, Graz, Austria, 1999.Google Scholar
  7. 7.
    A. Saalfeld, Joint triangulations and triangulation maps. Proc. 3rd Ann. ACM Sympos. Computational Geometry, Waterloo, Canada, 1987, 195–204.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Oswin Aichholzer
    • 1
  • Franz Aurenhammer
    • 1
  • Hannes Krasser
    • 1
  • Ferran Hurtado
    • 2
  1. 1.Institute for Theoretical Computer ScienceGraz University of TechnologyGrazAustria
  2. 2.Departament de Matematica Aplicada IIUniversitat Politecnica de CatalunyaBarcelonaSpain

Personalised recommendations