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Crossing Stars in Topological Graphs

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

Abstract

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.

Keywords

Common Vertex Topological Graph Geometric Graph Span Subgraph Common Endpoint 
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.

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References

  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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