On Graphs Supported by Line Sets

  • Vida Dujmović
  • William Evans
  • Stephen Kobourov
  • Giuseppe Liotta
  • Christophe Weibel
  • Stephen Wismath
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6502)


For a set S of n lines labeled from 1 to n, we say that S supports an n-vertex planar graph G if for every labeling from 1 to n of its vertices, G has a straight-line crossing-free drawing with each vertex drawn as a point on its associated line. It is known from previous work [4] that no set of n parallel lines supports all n-vertex planar graphs. We show that intersecting lines, even if they intersect at a common point, are more “powerful” than a set of parallel lines. In particular, we prove that every such set of lines supports outerpaths, lobsters, and squids, none of which are supported by any set of parallel lines. On the negative side, we prove that no set of n lines that intersect in a common point supports all n-vertex planar graphs. Finally, we show that there exists a set of n lines in general position that does not support all n-vertex planar graphs.


  1. 1.
    Badent, M., Giacomo, E.D., Liotta, G.: Drawing colored graphs on colored points. Theor. Comput. Sci. 408(2-3), 129–142 (2008)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bose, P.: On embedding an outer-planar graph in a point set. Computational Geometry: Theory and Applications 23(3), 303–312 (2002)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    de Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10(1), 41–51 (1990)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Estrella-Balderrama, A., Fowler, J.J., Kobourov, S.G.: Characterization of unlabeled level planar trees. Computational Geometry: Theory and Applications 42(7), 704–721 (2009)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Fowler, J.J., Kobourov, S.G.: Characterization of unlabeled level planar graphs. In: Hong, S.-H., Nishizeki, T., Quan, W. (eds.) GD 2007. LNCS, vol. 4875, Springer, Heidelberg (2008)Google Scholar
  6. 6.
    Gritzmann, P., Mohar, B., Pach, J., Pollack, R.: Embedding a planar triangulation with vertices at specified points. American Math. Monthly 98, 165–166 (1991)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Pach, J., Wenger, R.: Embedding planar graphs at fixed vertex locations. Graphs and Combinatorics 17, 717–728 (2001)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Rosenstiehl, P., Tarjan, R.E.: Rectilinear planar layouts and bipolar orientations of planar graphs. Discrete Computational Geometry 1(4), 343–353 (1986)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Vida Dujmović
    • 1
  • William Evans
    • 2
  • Stephen Kobourov
    • 3
  • Giuseppe Liotta
    • 4
  • Christophe Weibel
    • 5
  • Stephen Wismath
    • 6
  1. 1.School of Computer ScienceCarleton UniversityCanada
  2. 2.Dept. of Computer ScienceUniv. of British ColumbiaCanada
  3. 3.Dept. of Computer ScienceUniv. of ArizonaUSA
  4. 4.Dept. of Computer ScienceUniv. of PerugiaItaly
  5. 5.Dept. of MathematicsMcGill UniversityCanada
  6. 6.Dept. of Computer ScienceUniv. of LethbridgeCanada

Personalised recommendations