Crossing Stars in Topological Graphs

  • Gábor Tardos
  • Géza Tóth
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3742)


Let G be a graph without loops or multiple edges drawn in the plane. It is shown that, for any k, if G has at least C k n edges and n vertices, then it contains three sets of k edges, such that every edge in any of the sets crosses all edges in the other two sets. Furthermore, two of the three sets can be chosen such that all k edges in the set have a common vertex.


Common Vertex Topological Graph Geometric Graph Span Subgraph Common Endpoint 
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  1. 1.
    Agarwal, P.K., Aronov, B., Pach, J., Pollack, R., Sharir, M.: Quasi-planar graphs have a linear number of edges. Combinatorica 17, 1–9 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Pach, J.: Geometric graph theory. In: Lamb, J.D., Preece, D.A. (eds.) Surveys in Combinatorics. London Mathematical Society Lecture Notes, vol. 267, pp. 167–200. Cambridge University Press, Cambridge (1999)Google Scholar
  3. 3.
    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
  4. 4.
    Pach, J., Radoičić, R., Tóth, G.: A generalization of quasi-planarity. In: Pach, J. (ed.) Towards a Theory of Geometric Graphs. Contemporary Mathematics, vol. 342, pp. 177–183. AMS (2004)Google Scholar
  5. 5.
    Pach, J., Pinchasi, R., Sharir, M., Tóth, G.: Topological graphs with no large grids, Graphs and Combinatorics Special Issue dedicated to Victor Neumann-LaraGoogle Scholar
  6. 6.
    Pach, J., Pinchasi, R., Tardos, G., Tóth, G.: Geometric graphs with no self-intersecting path of length three. In: Goodrich, M.T., Kobourov, S.G. (eds.) GD 2002. LNCS, vol. 2528, pp. 295–311. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  7. 7.
    Pach, J., Radoičić, R., Tardos, G., Tóth, G.: Improving the Crossing Lemma by finding more crossings in sparse graphs. In: Proceedings of the 20th Annual Symposium on Computational Geometry (SoCG 2004), pp. 76–85 (2004)Google Scholar
  8. 8.
    Pach, J., Tóth, G.: Graphs drawn with few crossings per edge. Combinatorica 17, 427–439 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Schaefer, M., Stefankovič, D.: Decidability of string graphs. In: Proceedings of the 33rd Annual Symposium on the Theory of Computing (STOC 2001), pp. 241–246 (2001)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Gábor Tardos
    • 1
  • Géza Tóth
    • 1
  1. 1.Rényi InstituteHungarian Academy of SciencesBudapestHungary

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