, Volume 14, Issue 4, pp 327–363 | Cite as

The Complexity of Upward Drawings on Spheres

  • S. Mehdi Hashemi
  • Ivan Rival
  • Andrzej Kisielewicz


Although there is a linear time algorithm to decide whether an ordered set has an upward drawing on a surface topologically equivalence to a sphere, we shall prove that the decision problem whether an ordered set has an upward drawing on a sphere is NP-complete. The proof involves the investigation of the surface topology of ordered sets highlighting especially their saddle points. It echoes the recent, important result due to A. Garg and R. Tamassia (1995) that upward planarity testing is NP-complete, for which we give a new proof.

ordered set upward drawing sphere saddle point 


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  1. 1.
    Baker, K. A., Fishburn, P., and Roberts, F. S. (1971) Partial ordered sets of dimension 2, Networks 2, 11–28.Google Scholar
  2. 2.
    Bertolazzi, P., di Battista, G., Liotta, G., and Mannino, C. (1994) Upward drawings of triconnected digraphs, Algorithmica 12, 476–497.Google Scholar
  3. 3.
    Cormen, T. H., Leiserson, C. E., and Rivest, R. L. (1990) Introduction to Algorithms, The MIT Press, Cambridge, MA.Google Scholar
  4. 4.
    Czyżowicz, J., Pelc, A., and Rival, I. (1990) Planar ordered sets of width two, Math. Slovaca 40, 375–388.Google Scholar
  5. 5.
    di Battista, G., Liu, W. P., and Rival, I. (1990) Bipartite graphs, upward drawings, and planarity, Inform. Process. Lett. 36, 317–322.Google Scholar
  6. 6.
    di Battista, G. and Tamassia, R. (1988) Algorithms for plane representations of acyclic digraphs, Theoret. Comput. Sci. 61, 175–198.Google Scholar
  7. 7.
    Ewacha, K., Li, W., and Rival, I. (1991) Order, genus and diagram invariance, Order 8, 107–113.Google Scholar
  8. 8.
    Foldes, S., Rival, I., and Urrutia, J., Light source, obstructions, and spherical orders, Discrete Math. 102, 13–23.Google Scholar
  9. 9.
    Garey, M. R. and Johnson, D. S. (1979) Computers and Intractability: a Guide to the Theory of NP-completeness, Freeman, San Francisco, CA.Google Scholar
  10. 10.
    Garg, A. and Tamassia, R. (1994) On the computational complexity of upward and rectilinear planarity testing (Extended Abstract), Department of Computer Science, Brown University, Technical Report CS–94–10, 1–15.Google Scholar
  11. 11.
    Garg, A. and Tamassia, R. (1995) Upward planarity testing, Order 12(2), 109–133.Google Scholar
  12. 12.
    Glass, L. (1973) A combinatorial analog of the Poincaré index theorem, J. Combin. Theory 15, 264–268.Google Scholar
  13. 13.
    Hashemi, S. M. and Rival, I. (1994) Upward drawings to fit surfaces, in: V. Bouchitté and M. Morvan (eds.), Orders, Algorithms, and Applications (ORDAL '94), Lecture Notes in Computer Sciences 831, Springer-Verlag, Berlin, Heidelberg, New York, pp. 53–58.Google Scholar
  14. 14.
    Hashemi, S. M., Kisielewicz, A. and Rival, I. (1995) Upward drawing on planes and spheres, in: F. J. Brandenburg (ed.), Graph Drawing '95, Lecture Notes in Computer Science 1027, Springer-Verlag, Berlin, Heidelberg, New York, pp. 277–286.Google Scholar
  15. 15.
    Hopcroft, J. and Tarjan, R. E. (1979) Efficient planarity testing, J. Assoc. Comput. Mach. 21, 549–568.Google Scholar
  16. 16.
    Hutton, M. and Lubiw, A. (1991) Upward planar drawing of single source acyclic digraphs, in: Proc. 2nd ACM-SIAM Symp. Discrete Algorithms, San Francisco, pp. 203–211.Google Scholar
  17. 17.
    Kelly, D. (1987) Fundamentals of planar ordered sets, Discrete Math. 63, 197–216.Google Scholar
  18. 18.
    Kelly, D. and Rival, I. (1975) Planar lattices, Canad. J. Math. 27, 636–665.Google Scholar
  19. 19.
    Kisielewicz, A. (1994) Planar ordered sets and acyclic digraphs, Preprint.Google Scholar
  20. 20.
    Kisielewicz, A. and Rival, I. (1993) Every triangle-free planar graph has an upward planar drawing, Order 10, 1–16.Google Scholar
  21. 21.
    Musin, O. R., Rival, I., and Tarasov, S. P. (1993) Upward drawings on surfaces and the index of a critical point, Preprint.Google Scholar
  22. 22.
    Platt, C. R. (1976) Planar lattices and planar graphs, J. Combin. Theory Ser. B 21, 30–39.Google Scholar
  23. 23.
    Rival, I. (1993) Reading, drawing, and order, in: I. G. Rosenberg and G. Sabidussi (eds.), Algrebras and Orders, Kluwer Academic Publ., Dordrecht, pp. 359–404.Google Scholar

Copyright information

© Kluwer Academic Publishers 1997

Authors and Affiliations

  • S. Mehdi Hashemi
    • 1
  • Ivan Rival
    • 1
  • Andrzej Kisielewicz
    • 2
  1. 1.Department of MathematicsUniversity of OttawaOttawaCanada
  2. 2.Mathematical InstituteUniversity of WroclawWroclawPoland

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