Simultaneously Flippable Edges in Triangulations

  • Diane L. Souvaine
  • Csaba D. Tóth
  • Andrew Winslow
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7579)


Given a straight-line triangulation T, an edge e in T is flippable if e is adjacent to two triangles that form a convex quadrilateral. A set of edges E in T is simultaneously flippable if each edge is flippable and no two edges are adjacent to a common triangle. Intuitively, an edge is flippable if it may be replaced with the other diagonal of its quadrilateral without creating edge-edge intersections, and a set of edges is simultaneously flippable if they may be all be flipped without interferring with each other. We show that every straight-line triangulation on n vertices contains at least (n − 4)/5 simultaneously flippable edges. This bound is the best possible, and resolves an open problem by Galtier et al.


planar graph graph transformation geometry combinatorics 


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  1. 1.
    Bose, P., Hurtado, F.: Flips in planar graphs. Computational Geometry: Theory and Applications 42, 60–80 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Dumitrescu, A., Schulz, A., Sheffer, A., Tóth, C.D.: Bounds on the maximum multiplicity of some common geometric graphs. In: 28th International Symposium on Theoretical Aspects of Computer Science, pp. 637–648. Dagstuhl Publishing, Germany (2011)Google Scholar
  3. 3.
    Galtier, J., Hurtado, F., Noy, M., Pérennes, S., Urrutia, J.: Simultaneous edge flipping in triangulations. International Journal of Computational Geometry and Applications 13, 113–133 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Hoffmann, M., Sharir, M., Sheffer, A., Tóth, C.D., Welzl, E.: Counting Plane Graphs: Flippability and Its Applications. In: Dehne, F., Iacono, J., Sack, J.-R. (eds.) WADS 2011. LNCS, vol. 6844, pp. 524–535. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  5. 5.
    Hurtado, F., Noy, M., Urrutia, J.: Flipping edges in triangulations. Discrete and Compututational Geometry 22, 333–346 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Mucha, M., Sankowski, P.: Maximum matchings in planar graphs via Gaussian elimination. Algorithmica 45, 3–20 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Micali, S., Vazirani, V.V.: An \(O(\sqrt{|V|}\cdot|E|)\) algorithm for finding maximum matching in general graphs. In: 21st IEEE Symposium on Foundations of Computer Science, pp. 17–27. IEEE Press, New York (1980)Google Scholar
  8. 8.
    Urrutia, J.: Flipping edges in triangulations of point sets, polygons and maximal planar graphs. Invited talk, 1st Japanese Conference on Discrete and Computational Geometry, JCDCG 1997 (1997),

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Diane L. Souvaine
    • 1
  • Csaba D. Tóth
    • 2
  • Andrew Winslow
    • 1
  1. 1.Tufts UniversityMedfordUSA
  2. 2.University of CalgaryCalgaryCanada

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