k-Quasi-Planar Graphs

  • Andrew Suk
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7034)

Abstract

A topological graph is k-quasi-planar if it does not contain k pairwise crossing edges. A topological graph is simple if every pair of its edges intersect at most once (either at a vertex or at their intersection). In 1996, Pach, Shahrokhi, and Szegedy [16] showed that every n-vertex simple k-quasi-planar graph contains at most \(O\left(n(\log n)^{2k-4}\right)\) edges. This upper bound was recently improved (for large k) by Fox and Pach [8] to n(logn) O(logk). In this note, we show that all such graphs contain at most \((n\log^2n )2^{\alpha^{c_k}(n)}\) edges, where α(n) denotes the inverse Ackermann function and c k is a constant that depends only on k.

Keywords

Absolute Constant Topological Graph Annual Symposium Geometric Graph Crossing Edge 
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.

References

  1. 1.
    Ackerman, E.: On the Maximum Number of Edges in Topological Graphs with No Four Pairwise Crossing Edges. In: Proceedings of the Twenty-Second Annual Symposium on Computational Geometry, SCG 2006, pp. 259–263. ACM, New York (2006)CrossRefGoogle Scholar
  2. 2.
    Ackerman, E., Fox, J., Pach, J., Suk, A.: On Grids in Topological Graphs. In: Proceedings of the 25th Annual Symposium on Computational Geometry, SCG 2009, pp. 403–412. ACM, New York (2009)Google Scholar
  3. 3.
    Ackerman, E., Tardos, G.: Note: On the Maximum Number of Edges in Quasi-Planar Graphs. J. Comb. Theory Ser. A 114(3), 563–571 (2007)CrossRefMATHGoogle Scholar
  4. 4.
    Agarwal, P.K., Aronov, B., Pach, J., Pollack, R., Sharir, M.: Quasi-Planar Graphs Have a Linear Number of Edges. In: Brandenburg, F.J. (ed.) GD 1995. LNCS, vol. 1027, pp. 1–7. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  5. 5.
    Brass, P., Moser, W., Pach, J.: Research Problems in Discrete Geometry. Springer, Berlin (2005)MATHGoogle Scholar
  6. 6.
    Capoyleas, V., Pach, J.: A Turán-Type Theorem on Chords of a Convex Polygon. J. Combinatorial Theory, Series B 56, 9–15 (1992)CrossRefMATHGoogle Scholar
  7. 7.
    Dilworth, R.P.: A decomposition theorem for partially ordered sets. Annals of Math 51, 161–166 (1950)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Fox, J., Pach, J.: Coloring K k-Free Intersection Graphs of Geometric Objects in the Plane. In: Proceedings of the Twenty-Fourth Annual Symposium on Computational Geometry, SCG 2008, pp. 346–354. ACM, New York (2008)CrossRefGoogle Scholar
  9. 9.
    Fox, J., Pach, J., Tóth, C.: Intersection Patterns of Curves. Journal of the London Mathematical Society 83, 389–406 (2011)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Fulek, R., Suk, A.: Disjoint Crossing Families. In: EuroComb 2011 (2011, to appear)Google Scholar
  11. 11.
    Klazar, M.: A General Upper Bound in Extremal Theory of Sequences. Commentationes Mathematicae Universitatis Carolinae 33(4), 737–746 (1992)MathSciNetMATHGoogle Scholar
  12. 12.
    Klazar, M., Valtr, P.: Generalized Davenport-Schinzel Sequences. Combinatorica 14, 463–476 (1994)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Nivasch, G.: Improved Bounds and New Techniques for Davenport–Schinzel Sequences and Their Generalizations. J. ACM 57(3), 3, Article 17 (2010)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Pach, J., Radoičić, R., Tóth, G.: Relaxing Planarity for Topological Graphs. In: Akiyama, J., Kano, M. (eds.) JCDCG 2002. LNCS, vol. 2866, pp. 221–232. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  15. 15.
    Pach, J., Pinchasi, R., Sharir, M., Tóth, G.: Topological Graphs with No Large Grids. Graph. Comb. 21(3), 355–364 (2005)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Pach, J., Shahrokhi, F., Szegedy, M.: Applications of the Crossing Number. J. Graph Theory 22, 239–243 (1996)CrossRefMATHGoogle Scholar
  17. 17.
    Pettie, S.: Generalized Davenport-Schinzel Sequences and Their 0-1 Matrix Counterparts. J. Comb. Theory Ser. A 118(6), 1863–1895 (2011)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Pettie, S.: On the Structure and Composition of Forbidden Sequences, with Geometric Applications. In: Proceedings of the 27th Annual ACM Symposium on Computational Geometry, SCG 2011, pp. 370–379. ACM, New York (2011)Google Scholar
  19. 19.
    Tardos, G., Tóth, G.: Crossing Stars in Topological Graphs. SIAM J. Discret. Math. 21(3), 737–749 (2007)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Valtr, P.: On Geometric Graphs with No k Pairwise Parallel Edges. Discrete Comput. Geom. 19(3), 461–469 (1997)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Valtr, P.: Graph Drawings with No k Pairwise Crossing Edges. In: Di Battista, G. (ed.) GD 1997. LNCS, vol. 1353, pp. 205–218. Springer, Heidelberg (1997)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Andrew Suk
    • 1
  1. 1.School of Basic SciencesÉcole Polytechnique Fédérale de LausanneSwitzerland

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