Graph drawing with no k pairwise crossing edges

  • Pavel Valtr
Crossings and Planarity
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1353)


A geometric graph is a graph G = (V, E) drawn in the plane so that the vertex set V consists of points in general position and the edge set E consists of straight line segments between points of V. It is known that, for any fixed k, any geometric graph G on n vertices with no k pairwise crossing edges contains at most O(n log n) edges. In this paper we give a new, simpler proof of this bound, and show that the same bound holds also when the edges of G are represented by x-monotone curves (Jordan arcs).


  1. 1.
    R. Adamec, M. Klazar, and P. Valtr, Generalized Davenport-Schinzel sequences with linear upper bound, Discrete Math. 108 (1992), 219–229.CrossRefGoogle Scholar
  2. 2.
    P.K. Agarwal, B. Aronov, J. Pach, R. Pollack, and M. Sharir, Quasi-planar graphs have a linear number of edges, Graph Drawing (Passau, 1995), Lecture Notes in Comput. Sci. 1027 (1995), 1–7.Google Scholar
  3. 3.
    H. Davenport and A. Schinzel, A combinatorial problem connected with differential equations, Amer. J. Math. 87 (1965), 684–694.Google Scholar
  4. 4.
    R. Dilworth, A decomposition theorem for partially ordered sets, Annals of Mathematics 51 (1950), 161–166.Google Scholar
  5. 5.
    P. Erdős and G. Szekeres, A combinatorial problem in geometry, Compositio Math. 2 (1935), 463–470.Google Scholar
  6. 6.
    M. Klazar and P. Valtr, Generalized Davenport-Schinzel sequences, Combinatorica 14 (1994), 463–476.CrossRefGoogle Scholar
  7. 7.
    J. Pach, Notes on geometric graph theory, in: Discrete and Computational Geometry: Papers from DIMACS Special Year, DIMACS Series, Vol. 6, AMS, Providence, RI, 1991, pp. 273–285.Google Scholar
  8. 8.
    J. Pach and P.K. Agarwal, Combinatorial Geometry, Wiley Interscience, New York (1995).Google Scholar
  9. 9.
    J. Pach, F. Sharokhi, and M. Szegedy, Applications of the crossing number, Algorithmica 16 (1996), 111–117.Google Scholar
  10. 10.
    J. Pach and J. Tlörőcsik, Some geometric applications of Dilworth's theorem, Discrete Comput. Geom. 12 (1994), 1–7.Google Scholar
  11. 11.
    F. P. Ramsey, On a problem of formal logic, Proc. Lond. Math. Soc., II. Ser., 30 (1930),264–286.Google Scholar
  12. 12.
    P. Valtr, On geometric graphs with no k pairwise parallel edges, to appear in Discrete and Computational Geometry.Google Scholar

Copyright information

© Springer-Verlag 1997

Authors and Affiliations

  • Pavel Valtr
    • 1
    • 2
  1. 1.Department of Applied MathematicsCharles UniversityPraha 1Czech Republic
  2. 2.DIMACS CenterRutgers UniversityPiscatawayUSA

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