On the Size of Planarly Connected Crossing Graphs

  • Eyal Ackerman
  • Balázs KeszeghEmail author
  • Mate Vizer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9801)


We prove that if an n-vertex graph G can be drawn in the plane such that each pair of crossing edges is independent and there is a crossing-free edge that connects their endpoints, then G has O(n) edges. Graphs that admit such drawings are related to quasi-planar graphs and to maximal 1-planar and fan-planar graphs.


Planar graphs Crossing edges Crossing-free edge Fan-planar graphs 1-planar graphs 



We thank Géza Tóth for his permission to include his construction for a lower bound on the size of a PCC graph in this paper. We also thank an anonymous referee for pointing out an error in an earlier version of this paper.

Most of this work was done during a visit of the first author to the Rényi Institute that was partially supported by the National Research, Development and Innovation Office – NKFIH under the grant PD 108406 and by the ERC Advanced Research Grant no. 267165 (DISCONV). The second author was supported by the National Research, Development and Innovation Office – NKFIH under the grant PD 108406 and K 116769 and by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences. The third author was supported by Development and Innovation Office – NKFIH under the grant SNN 116095.


  1. 1.
    Ackerman, E.: On the maximum number of edges in topological graphs with no four pairwise crossing edges. Discrete Computat. Geom. 41(3), 365–375 (2009). MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ackerman, E., Fox, J., Pach, J., Suk, A.: On grids in topological graphs. Comput. Geom. 47(7), 710–723 (2014). MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ackerman, E., Fulek, R., Tóth, C.D.: Graphs that admit polyline drawings with few crossing angles. SIAM J. Discrete Math. 26(1), 305–320 (2012). MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ackerman, E., Tardos, G.: On the maximum number of edges in quasi-planar graphs. J. Comb. Theory Ser. A 114(3), 563–571 (2007). MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Agarwal, P.K., Aronov, B., Pach, J., Pollack, R., Sharir, M.: Quasi-planar graphs have a linear number of edges. Combinatorica 17(1), 1–9 (1997). MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Brass, P., Moser, W.O.J., Pach, J.: Research Problems in Discrete Geometry. Springer, New York (2005)zbMATHGoogle Scholar
  7. 7.
    Capoyleas, V., Pach, J.: A Turán-type theorem on chords of a convex polygon. J. Comb. Theory Ser. B 56(1), 9–15 (1992). MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chojnacki, C.: Über wesentlich unplättbare Kurven im dreidimensionalen Raume. Fundamenta Mathematicae 23(1), 135–142 (1934)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Fox, J., Pach, J.: Coloring \({K}_k\)-free intersection graphs of geometric objects in the plane. Eur. J. Comb. 33(5), 853–866 (2012). MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Fox, J., Pach, J.: Applications of a new separator theorem for string graphs. Comb. Probab. Comput. 23(1), 66–74 (2014). MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kaufmann, M., Ueckerdt, T.: The density of fan-planar graphs. CoRR abs/1403.6184 (2014).
  12. 12.
    Pach, J.: Notes on geometric graph theory. In: Goodman, J., Pollack, R., Steiger, W. (eds.) Discrete and Computational Geometry: Papers from DIMACS special year, DIMACS series, vol. 6, pp. 273–285. AMS, Providence (1991)Google Scholar
  13. 13.
    Pach, J., Radoičić, R., Tóth, G.: Relaxing planarity for topological graphs. In: Gőri, E., Katona, G.O., Lovász, L. (eds.) More Graphs, Sets and Numbers, Bolyai Society Mathematical Studies, vol. 15, pp. 285–300. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  14. 14.
    Pach, J., Tóth, G.: Graphs drawn with few crossings per edge. Combinatorica 17(3), 427–439 (1997). MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Pelsmajer, M.J., Schaefer, M., Štefankovič, D.: Removing even crossings. J. Comb. Theory Ser. B 97(4), 489–500 (2007). MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Suk, A., Walczak, B.: New bounds on the maximum number of edges in k-quasi-planar graphs. Comput. Geom. 50, 24–33 (2015). MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Tóth, G.: Private communication (2015)Google Scholar
  18. 18.
    Tutte, W.: Toward a theory of crossing numbers. J. Comb. Theory 8(1), 45–53 (1970). MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  1. 1.Department of Mathematics, Physics, and Computer ScienceUniversity of Haifa at OranimTivonIsrael
  2. 2.Alfréd Rényi Institute of MathematicsHungarian Academy of SciencesBudapestHungary

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