Reconfiguring Triangulations with Edge Flips and Point Moves

  • Greg Aloupis
  • Prosenjit Bose
  • Pat Morin
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3383)


We examine reconfigurations between triangulations and near-triangulations of point sets, and give new bounds on the number of point moves and edge flips sufficient for any reconfiguration. We show that with O(nlog n) edge flips and point moves, we can transform any geometric near-triangulation on n points to any other geometric near-triangulation on n possibly different points. This improves the previously known bound of O(n2) edge flips and point moves.


  1. 1.
    Abellanas, M., Bose, P., Garcia, A., Hurtado, F., Ramos, P., Rivera-Campo, E., Tejel, J.: On local transformations in plane geometric graphs embedded on small grids. In: Proceedings of the International Workshop on Computational Geometry and Applications (CGA), vol. 2, pp. 22–31 (2004)Google Scholar
  2. 2.
    Bose, P., Czyzowicz, J., Gao, Z., Morin, P., Wood, D.: Parallel diagonal flips in plane triangulations. Tech. Rep. TR-2003-05, School of Computer Science, Carleton University, Ottawa, Canada (2003)Google Scholar
  3. 3.
    Brunet, R., Nakamoto, A., Negami, S.: Diagonal flips of triangulations on closed surfaces preserving specified properties. J. Combin. Theory Ser. B 68(2), 295–309 (1996)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Cortés, C., Grima, C., Marquez, A., Nakamoto, A.: Diagonal flips in outertriangulations on closed surfaces. Discrete Math. 254(1-3), 63–74 (2002)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Cortés, C., Nakamoto, A.: Diagonal flips in outer-torus triangulations. Discrete Math. 216(1-3), 71–83 (2000)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Galtier, J., Hurtado, F., Noy, M., Pérennes, S., Urrutia, J.: Simultaneous edge flipping in triangulations. Internat. J. Comput. Geom. Appl. 13(2), 113–133 (2003)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Gao, Z., Urrutia, J., Wang, J.: Diagonal flips in labelled planar triangulations. Graphs Combin. 17(4), 647–657 (2001)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Hurtado, F., Noy, M.: Graph of triangulations of a convex polygon and tree of triangulations. Comput. Geom. 13(3), 179–188 (1999)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Hurtado, F., Noy, M., Urrutia, J.: Flipping edges in triangulations. Discrete Comput. Geom. 22(3), 333–346 (1999)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Komuro, H.: The diagonal flips of triangulations on the sphere. Yokohama Math. J. 44(2), 115–122 (1997)MATHMathSciNetGoogle Scholar
  11. 11.
    Komuro, H., Nakamoto, A., Negami, S.: Diagonal flips in triangulations on closed surfaces with minimum degree at least 4. J. Combin. Theory Ser. B 76(1), 68–92 (1999)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Lawson, C.: Software for c1 surface interpolation. In: Rice, J. (ed.) Mathematical Software III, pp. 161–194. Academic Press, New York (1977)Google Scholar
  13. 13.
    Meisters, G.: Polygons have ears. American Mathematical Monthly 82, 648–651 (1975)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Nakamoto, A., Negami, S.: Diagonal flips in graphs on closed surfaces with specified face size distributions. Yokohama Math. J. 49(2), 171–180 (2002)MATHMathSciNetGoogle Scholar
  15. 15.
    Negami, S.: Diagonal flips in triangulations of surfaces. Discrete Math. 135(1-3), 225–232 (1994)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Negami, S.: Diagonal flips in triangulations on closed surfaces, estimating upper bounds. Yokohama Math. J. 45(2), 113–124 (1998)MATHMathSciNetGoogle Scholar
  17. 17.
    Negami, S.: Diagonal flips of triangulations on surfaces, a survey. Yokohama Math. J. 47, 1–40 (1999)MATHMathSciNetGoogle Scholar
  18. 18.
    Negami, S., Nakamoto, A.: Diagonal transformations of graphs on closed surfaces. Sci. Rep. Yokohama Nat. Univ. Sect. I Math. Phys. Chem. 40, 71–97 (1993)MathSciNetGoogle Scholar
  19. 19.
    Wagner, K.: Bemerkung zum Vierfarbenproblem. Jber. Deutsch. Math.-Verein. 46, 26–32 (1936)Google Scholar
  20. 20.
    Watanabe, T., Negami, S.: Diagonal flips in pseudo-triangulations on closed surfaces without loops. Yokohama Math. J. 47, 213–223 (1999)MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Greg Aloupis
    • 1
  • Prosenjit Bose
    • 2
  • Pat Morin
    • 2
  1. 1.School of Computer ScienceMcGill University 
  2. 2.School of Computer ScienceCarleton University 

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