Upper Bound Constructions for Untangling Planar Geometric Graphs

  • Javier Cano
  • Csaba D. Tóth
  • Jorge Urrutia
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7034)


For every n ∈ ℕ, there is a straight-line drawing D n of a planar graph on n vertices such that in any crossing-free straight-line drawing of the graph, at most O(n .4982) vertices lie at the same position as in D n . This improves on an earlier bound of \(O(\sqrt{n})\) by Goaoc et al. [6].


Planar Graph Jordan Curve Outer Face Interior Vertex Marked Vertex 
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.


  1. 1.
    Arkin, E.M., Held, M., Mitchell, J.S.B., Skiena, S.: Hamiltonian triangulations for fast rendering. The Visual Computer 12(9), 429–444 (1996)CrossRefGoogle Scholar
  2. 2.
    Bose, P., Dujmovic, V., Hurtado, F., Langerman, S., Morin, P., Wood, D.R.: A polynomial bound for untangling geometric planar graphs. Discrete Comput. Geom. 42(4), 570–585 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cibulka, J.: Untangling polygons and graphs. Discrete Comput. Geom. 43, 402–411 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Erdős, P., Szekeres, G.: A combinatorial problem in geometry. Compositio Mathematica 2, 463–470 (1935)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Fáry, I.: On straight line representation of planar graphs. Acta Univ. Szeged, Acta Sci. Math. 11, 229–233 (1948)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Goaoc, X., Kratochvíl, J., Okamoto, Y., Shin, C.S., Spillner, A., Wolff, A.: Untangling a planar graph. Discrete Comput. Geom. 42(4), 542–569 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Holton, D.A., McKay, B.D.: The smallest non-Hamiltonian 3-connected cubic planar graphs have 38 vertices. J. Combin. Theory Ser. B 45(3), 305–319 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Kang, M., Pikhurko, O., Ravsky, A., Schacht, M., Verbitsky, O.: Untangling planar graphs from a specified vertex position—Hard cases. Discrete Appl. Math. 159(8), 789–799 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Pach, J., Tardos, G.: Untangling a polygon. Discrete Comput. Geom. 28(4), 585–592 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Tait, P.G.: Listing’s Topologie. Philosophical Magazine 17, 30–46 (1884)CrossRefGoogle Scholar
  11. 11.
    Tutte, W.T.: On Hamiltonian circuits. J. LMS 21(2), 98–101 (1946)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Javier Cano
    • 1
  • Csaba D. Tóth
    • 2
  • Jorge Urrutia
    • 3
  1. 1.Posgrado en Ciencia e Ingeniería de la ComputaciónUniversidad Nacional Autónoma de MéxicoMéxico
  2. 2.Department of Math.University of CalgaryCanada
  3. 3.Instituto de MatemáticasUniversidad Nacional Autónoma de MéxicoMéxico

Personalised recommendations