On Collinear Sets in Straight-Line Drawings

  • Alexander Ravsky
  • Oleg Verbitsky
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6986)


We consider straight-line drawings of a planar graph G with possible edge crossings. The untangling problem is to eliminate all edge crossings by moving as few vertices as possible to new positions. Let fix G denote the maximum number of vertices that can be left fixed in the worst case among all drawings of G. In the allocation problem, we are given a planar graph G on n vertices together with an n-point set X in the plane and have to draw G without edge crossings so that as many vertices as possible are located in X. Let fit G denote the maximum number of points fitting this purpose in the worst case among all n-point sets X. As fix G ≤ fit G, we are interested in upper bounds for the latter and lower bounds for the former parameter.

For any ε > 0, we construct an infinite sequence of graphs with fit G = O(n σ + ε ), where σ < 0.99 is a known graph-theoretic constant, namely the shortness exponent for the class of cubic polyhedral graphs. On the other hand, we prove that \(fix G\ge\sqrt{n/30}\) for any graph G of tree-width at most 2. This extends the lower bound obtained by Goaoc et al. [Discrete and Computational Geometry 42:542–569 (2009)] for outerplanar graphs. Our results are based on estimating the maximum number of vertices that can be put on a line in a straight-line crossing-free drawing of a given planar graph.


Planar Graph Outer Face Outerplanar Graph Plane Embedding Edge Crossing 
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.
    Bose, P., Dujmovic, V., Hurtado, F., Langerman, S., Morin, P., Wood, D.R.: A polynomial bound for untangling geometric planar graphs. Discrete and Computational Geometry 42, 570–585 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Cibulka, J.: Untangling polygons and graphs. Discrete and Computational Geometry 43, 402–411 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Felsner, S., Liotta, G., Wismath, S.K.: Straight-line drawings on restricted integer grids in two and three dimensions. J. Graph Algorithms Appl. 7, 363–398 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Flajolet, P., Noy, M.: Analytic combinatorics of non-crossing configurations. Discrete Mathematics 204, 203–229 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    García, A., Hurtado, F., Huemer, C., Tejel, J., Valtr, P.: On triconnected and cubic plane graphs on given point sets. Comput. Geom. 42, 913–922 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Giménez, O., Noy, M.: Counting planar graphs and related families of graphs. In: Surveys in Combinatorics 2009, pp. 169–210. Cambridge University Press, Cambridge (2009)CrossRefGoogle Scholar
  7. 7.
    Goaoc, X., Kratochvíl, J., Okamoto, Y., Shin, C.S., Spillner, A., Wolff, A.: Untangling a planar graph. Discrete and Computational Geometry 42, 542–569 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gritzmann, P., Mohar, B., Pach, J., Pollack, R.: Embedding a planar triangulation with vertices at specified points. Amer. Math. Monthly 98, 165–166 (1991)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Grünbaum, B., Walther, H.: Shortness exponents of families of graphs. J. Combin. Theory A 14, 364–385 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Jackson, B.: Longest cycles in 3-connected cubic graphs. J. Combin. Theory B 41, 17–26 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kang, M., Pikhurko, O., Ravsky, A., Schacht, M., Verbitsky, O.: Untangling planar graphs from a specified vertex position — Hard cases. Discrete Applied Mathematics 159, 789–799 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Mohar, B., Thomassen, C.: Graphs on surfaces. The John Hopkins University Press, Baltimore (2001)zbMATHGoogle Scholar
  13. 13.
    Pach, J., Tardos, G.: Untangling a polygon. Discrete and Computational Geometry 28, 585–592 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Ravsky, A., Verbitsky, O.: On collinear sets in straight-line drawings. E-print (2011),
  15. 15.
    Verbitsky, O.: On the obfuscation complexity of planar graphs. Theoretical Computer Science 396, 294–300 (2008)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Alexander Ravsky
    • 1
  • Oleg Verbitsky
    • 1
  1. 1.Institute for Applied Problems of Mechanics and MathematicsLvivUkraine

Personalised recommendations