On the Number of Edges of Fan-Crossing Free Graphs

  • Otfried Cheong
  • Sariel Har-Peled
  • Heuna Kim
  • Hyo-Sil Kim
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8283)

Abstract

A graph drawn in the plane with n vertices is fan-crossing free if there is no triple of edges e,f and g, such that e and f have a common endpoint and g crosses both e and f. We prove a tight bound of 4n − 9 on the maximum number of edges of such a graph for a straight-edge drawing. The bound is 4n − 8 if the edges are Jordan curves. We also discuss generalizations to monotone graph properties.

Keywords

graph theory graph drawing planar graph extremal graph 

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References

  1. 1.
    Ackerman, E.: On the maximum number of edges in topological graphs with no four pairwise crossing edges. Discrete & Computational Geometry 41(3), 365–375 (2009)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    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)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Argyriou, E.N., Bekos, M.A., Symvonis, A.: The straight-line RAC drawing problem is NP-hard. J. Graph Algorithms Appl. 16(2), 569–597 (2012)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Didimo, W.: Density of straight-line 1-planar graph drawings. Inf. Process. Lett. 113(7), 236–240 (2013)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Didimo, W., Eades, P., Liotta, G.: Drawing graphs with right angle crossings. Theo. Comp. Sci. 412(39), 5156–5166 (2011)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Eades, P., Liotta, G.: Right angle crossing graphs and 1-planarity. Discrete Applied Mathematics 161(7-8), 961–969 (2013)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Fox, J., Pach, J., Suk, A.: The number of edges in k-quasi-planar graphs. SIAM J. Discrete Math. 27(1), 550–561 (2013)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Hong, S.-H., Eades, P., Liotta, G., Poon, S.-H.: Fáry’s theorem for 1-planar graphs. In: Gudmundsson, J., Mestre, J., Viglas, T. (eds.) COCOON 2012. LNCS, vol. 7434, pp. 335–346. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  9. 9.
    Korzhik, V.P., Mohar, B.: Minimal obstructions for 1-immersions and hardness of 1-planarity testing. Journal of Graph Theory 72(1), 30–71 (2013)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Pach, J., Pinchasi, R., Sharir, M., Tóth, G.: Topological graphs with no large grids. Graphs and Combinatorics 21, 355–364 (2005)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    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
  12. 12.
    Pach, J., Shahrokhi, F., Szegedy, M.: Applications of the crossing number. In: Proc. 10th Annu. ACM Sympos. Comput. Geom., pp. 198–202 (1994)Google Scholar
  13. 13.
    Pach, J., Tóth, G.: Graphs drawn with few crossings per edge. Combinatorica 17, 427–439 (1997)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Otfried Cheong
    • 1
  • Sariel Har-Peled
    • 2
  • Heuna Kim
    • 3
  • Hyo-Sil Kim
    • 4
  1. 1.Department of Computer ScienceKAISTDaejeonKorea
  2. 2.Department of Computer ScienceUniversity of IllinoisUrbanaUSA
  3. 3.Freie Universität BerlinBerlinGermany
  4. 4.Department of Computer Science and EngineeringPOSTECHPohangKorea

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