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 Dn 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 Dn. This improves on an earlier bound of \(O(\sqrt{n})\) by Goaoc et al. [6].


  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)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Cibulka, J.: Untangling polygons and graphs. Discrete Comput. Geom. 43, 402–411 (2010)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Erdős, P., Szekeres, G.: A combinatorial problem in geometry. Compositio Mathematica 2, 463–470 (1935)MathSciNetMATHGoogle Scholar
  5. 5.
    Fáry, I.: On straight line representation of planar graphs. Acta Univ. Szeged, Acta Sci. Math. 11, 229–233 (1948)MathSciNetMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Pach, J., Tardos, G.: Untangling a polygon. Discrete Comput. Geom. 28(4), 585–592 (2002)MathSciNetCrossRefMATHGoogle 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)MathSciNetMATHGoogle 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