New Exact Results and Bounds for Bipartite Crossing Numbers of Meshes

  • Matthew C. Newton
  • Ondrej Sýkora
  • Martin Užovič
  • Imrich Vrt’o
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3383)

Abstract

The bipartite crossing number of a bipartite graph is the minimum number of crossings of edges when the partitions are placed on two parallel lines and edges are drawn as straight line segments between the lines. We prove exact results, asymtotics and new upper bounds for the bipartite crossing numbers of 2-dimensional mesh graphs. We especially show that bcr(P6× Pn)=35n–47, for n≥ 7.

References

  1. 1.
    Chung, F.R.K.: A conjectured minimum valuation tree. SIAM Review 20, 601–604 (1978)CrossRefGoogle Scholar
  2. 2.
    Demetrescu, C., Finocchi, I.: Removing cycles for minimizing crossings. ACM Journal of Experimental Algorithms 6 (2001)Google Scholar
  3. 3.
    Di Battista, J., Eades, P., Tamassia, R., Tollis, I.G.: Graph Drawing. Algorithms for the Visualization of Graphs. Prentice-Hall, Englewood Cliffs (1999)MATHGoogle Scholar
  4. 4.
    Erdös, P., Guy, R.P.: Crossing number problems. American Mathematical Monthly 80, 52–58 (1973)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Eades, P., Wormald, N.: Edge crossings in drawings of bipartite graphs. Algorithmica 11, 379–403 (1994)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Garey, M.R., Johnson, D.S.: Crossing number is NP-complete. SIAM Journal on Algebraic and Discrete Methods 4, 312–316 (1983)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Harary, F.: Determinants, permanents and bipartite graphs. Mathematical Magazine 42, 146–148 (1969)MATHCrossRefGoogle Scholar
  8. 8.
    Harary, F., Schwenk, A.: A new crossing number for bipartite graphs. Utilitas Mathematica 1, 203–209 (1972)MATHMathSciNetGoogle Scholar
  9. 9.
    Jünger, M., Mutzel, P.: 2-layer straight line crossing minimization: performance of exact and heuristic algorithms. Journal of Graph Algorithms and Applications 1, 1–25 (1997)MathSciNetGoogle Scholar
  10. 10.
    Leighton, F.T.: Complexity Issues in VLSI. MIT Press, Cambridge (1983)Google Scholar
  11. 11.
    Martí, R.: A tabu search algorithm for the bipartite drawing problem. European Journal of Operational Research 106, 558–569 (1998)MATHCrossRefGoogle Scholar
  12. 12.
    Matuszewski, C., Schönfeld, R., Molitor, P.: Using sifting for k-layer crossing minimization. In: Kratochvíl, J. (ed.) GD 1999. LNCS, vol. 1731, pp. 217–224. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  13. 13.
    May, M., Szkatula, K.: On the bipartite crossing number. Control and Cybernetics 17, 85–98 (1988)MATHGoogle Scholar
  14. 14.
    Newton, M., Sýkora, O., Vrt’o, I.: Two new heuristics for two-sided bipartite graph drawings. In: Goodrich, M.T., Kobourov, S.G. (eds.) GD 2002. LNCS, vol. 2528, pp. 312–319. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  15. 15.
    Pach, J., Agarwal, P.K.: Combinatorial Geometry. Wiley and Sons, New York (1995)MATHGoogle Scholar
  16. 16.
    Odenthal, T.: Personal communication (2002)Google Scholar
  17. 17.
    Sarrafzadeh, M.: An Introduction to VLSI Physical Design. McGraw-Hill, New York (1995)Google Scholar
  18. 18.
    Shahrokhi, F., Sýkora, O., Székely, L.A., Vrt’o, I.: On the bipartite drawings and the linear arrangement problem. In: Rau-Chaplin, A., Dehne, F., Sack, J.-R., Tamassia, R. (eds.) WADS 1997. LNCS, vol. 1272, pp. 55–68. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  19. 19.
    Shahrokhi, F., Sýkora, O., Székely, L.A., Vrt’o, I.: A new lower bound for the bipartite crossing number with applications. Theoretical Computer Science 245, 281–294 (2000)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Spinrad, J., Brandstädt, A., Stewart, L.: Bipartite permutation graphs. Discrete Applied Mathematics 19, 279–292 (1987)CrossRefGoogle Scholar
  21. 21.
    Sugiyama, K., Tagawa, S., Toda, M.: Methods for visual understanding of hierarchical systems structures. IEEE Transactions on Systems, Man and Cybernetics 11, 109–125 (1981)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Warfield, J.: Crossing theory and hierarchy mapping. IEEE Transactions on Systems. Man and Cybernetics 7, 502–523 (1977)MathSciNetGoogle Scholar
  23. 23.
    Watkins, M.E.: A special crossing number for bipartite graphs: a research problem. Annals of New York Academy Sciences 175, 405–410 (1970)MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Matthew C. Newton
    • 1
  • Ondrej Sýkora
    • 1
  • Martin Užovič
    • 2
  • Imrich Vrt’o
    • 3
  1. 1.Department of Computer ScienceLoughborough UniversityLoughboroughUnited Kingdom
  2. 2.Department of Computer ScienceComenius UniversityMlynská dolinaSlovak Republic
  3. 3.Department of InformaticsInstitute of Mathematics, Slovak Academy of SciencesBratislavaSlovak Republic

Personalised recommendations