Crossing Number of Graphs with Rotation Systems

  • Michael J. Pelsmajer
  • Marcus Schaefer
  • Daniel Štefankovič
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4875)


We show that computing the crossing number of a graph with a given rotation system is NP-complete. This result leads to a new and much simpler proof of Hliněný’s result, that computing the crossing number of a cubic graph (without rotation system) is NP-complete. We also investigate the special case of multigraphs with rotation systems on a fixed number k of vertices. For k = 1 and k = 2 the crossing number can be computed in polynomial time and approximated to within a factor of 2 in linear time. For larger k we show how to approximate the crossing number to within a factor of \({k+4\choose 4}/5\) in time O(m k + 2) on a graph with m edges.


crossing number computational complexity computational geometry 


  1. 1.
    Archdeacon, D.: Problems in topological graph theory (accessed Septmeber 15, 2006),
  2. 2.
    Bokal, D., Fijavž, G., Mohar, B.: Minor-monotone crossing number. In: Felsner, S. (ed.) EuroComb 2005. Discrete Mathematics and Theoretical Computer Science, vol. AE, pp. 123–128 (2005)Google Scholar
  3. 3.
    Buchheim, C., Jünger, M., Menze, A., Percan, M.: Directed crossing minimization. Technical report, Zentrum für Angewandte Informatik Köln, Lehrstuhl Jünger (August 2005)Google Scholar
  4. 4.
    Diaconis, P., Graham, R.L.: Spearman’s footrule as a measure of disarray. J. Roy. Statist. Soc. Ser. B 39(2), 262–268 (1977)MATHMathSciNetGoogle Scholar
  5. 5.
    Garey, M., Johnson, D.: Crossing number is NP-complete. SIAM Journal on Algebraic and Discrete Methods 4, 312–316 (1983)MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Hliněný, P.: Crossing number is hard for cubic graphs. J. Combin. Theory Ser. B 96(4), 455–471 (2006)CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Kedem, Z.M., Fuchs, H.: On finding several shortest paths in certain graphs. In: 18th Allerton Conference, pp. 677–686 (1980)Google Scholar
  8. 8.
    Knuth, D.E.: The art of computer programming. In: Sorting and searching, Addison-Wesley Series in Computer Science and Information Processing, vol. 3, Addison-Wesley Publishing Co., Reading (1973)Google Scholar
  9. 9.
    Lowrance, R., Wagner, R.A.: An extension of the string-to-string correction problem. J. Assoc. Comput. Mach. 22, 177–183 (1975)MATHMathSciNetGoogle Scholar
  10. 10.
    Maes, M.: On a cyclic string-to-string correction problem. Inform. Process. Lett. 35(2), 73–78 (1990)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Marzal, A., Barrachina, S.: Speeding up the computation of the edit distance for cyclic strings. In: International Conference on Pattern Recognition, pp. 891–894 (2000)Google Scholar
  12. 12.
    Štefankovič, D., Pelsmajer, M.J., Schaefer, M.: Removing even crossings. In: Felsner, S. (ed.) EuroComb 2005, DMTCS Proceedings. Discrete Mathematics and Theoretical Computer Science, vol. AE, pp. 105–110 (2005)Google Scholar
  13. 13.
    Pelsmajer, M.J., Schaefer, M., Štefankovič, D.: Odd crossing number is not crossing number. In: Healy, P., Nikolov, N.S. (eds.) GD 2005. LNCS, vol. 3843, pp. 386–396. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  14. 14.
    Pelsmajer, M.J., Schaefer, M., Štefankovič, D.: Removing even crossings. J. Combin. Theory Ser. B (to appear)Google Scholar
  15. 15.
    Wagner, R.A.: On the complexity of the extended string-to-string correction problem. In: Robert, A. (ed.) Seventh Annual ACM Symposium on Theory of Computing, Albuquerque, N.M., Assoc. Comput. Mach., New York, pp. 218–223 (1975)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Michael J. Pelsmajer
    • 1
  • Marcus Schaefer
    • 2
  • Daniel Štefankovič
    • 3
  1. 1.Illinois Institute of TechnologyChicagoUSA
  2. 2.DePaul UniversityChicagoUSA
  3. 3.University of RochesterRochesterUSA

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