Crossing and Weighted Crossing Number of Near-Planar Graphs

  • Sergio Cabello
  • Bojan Mohar
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5417)


A nonplanar graph G is near-planar if it contains an edge e such that G − e is planar. The problem of determining the crossing number of a near-planar graph is exhibited from different combinatorial viewpoints. On the one hand, we develop min-max formulas involving efficiently computable lower and upper bounds. These min-max results are the first of their kind in the study of crossing numbers and improve the approximation factor for the approximation algorithm given by Hliněný and Salazar (Graph Drawing GD 2006). On the other hand, we show that it is NP-hard to compute a weighted version of the crossing number for near-planar graphs.


  1. 1.
    Bhatt, S.N., Leighton, F.T.: A framework for solving VLSI graph layout problems. J. Comput. System Sci. 28(2), 300–343 (1984)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Bokal, D., Fijavž, G., Mohar, B.: The minor crossing number. SIAM J. Discret. Math. 20(2), 344–356 (2006)CrossRefzbMATHGoogle Scholar
  3. 3.
    Garey, M.R., Johnson, D.S.: Crossing number is NP-complete. SIAM J. Alg. Discr. Meth. 4, 312–316 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York (1979)zbMATHGoogle Scholar
  5. 5.
    Gutwenger, C., Mutzel, P., Weiskircher, R.: Inserting an edge into a planar graph. Algorithmica 41, 289–308 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Hliněný, P., Salazar, G.: On the crossing number of almost planar graphs. In: Kaufmann, M., Wagner, D. (eds.) GD 2006. LNCS, vol. 4372, pp. 162–173. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  7. 7.
    Juvan, M., Marinček, J., Mohar, B.: Elimination of local bridges. Math. Slovaca 47, 85–92 (1997)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Leighton, F.T.: Complexity issues in VLSI. MIT Press, MA (1983)Google Scholar
  9. 9.
    Leighton, F.T.: New lower bound techniques for vlsi. Math. Systems Theory 17, 47–70 (1984)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Liebers, A.: Planarizing graphs—a survey and annotated bibliography. J. Graph Algorithms Appl. 5, 74pp. (2001)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Mishra, B., Tarjan, R.E.: A linear-time algorithm for finding an ambitus. Algorithmica 7(5&6), 521–554 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Mohar, B., Thomassen, C.: Graphs on Surfaces. Johns Hopkins University Press, Baltimore (2001)zbMATHGoogle Scholar
  13. 13.
    Mohar, B.: On the crossing number of almost planar graphs. Informatica 30, 301–303 (2006)zbMATHMathSciNetGoogle Scholar
  14. 14.
    Riskin, A.: The crossing number of a cubic plane polyhedral map plus an edge. Studia Sci. Math. Hungar. 31, 405–413 (1996)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Shahrokhi, F., Sýkora, O., Székely, L.A., Vrt’o, I.: Crossing numbers: bounds and applications. In: Barany, I., Böröczky, K. (eds.) Intuitive geometry (Budapest, 1995). Bolyai Society Mathematical Studies, vol. 6, pp. 179–206. Akademia Kiado (1997)Google Scholar
  16. 16.
    Székely, L.A.: A successful concept for measuring non-planarity of graphs: The crossing number. Discrete Math. 276, 331–352 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Tutte, W.T.: Separation of vertices by a circuit. Discrete Math. 12, 173–184 (1975)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Vrt’o, I.: Crossing number of graphs: A bibliography,

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Sergio Cabello
    • 1
  • Bojan Mohar
    • 2
  1. 1.Department of Mathematics, FMFUniversity of LjubljanaSlovenia
  2. 2.Department of MathematicsSimon Fraser UniversityBurnabyCanada

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