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A Parameterized Algorithm for Upward Planarity Testing

(Extended Abstract)
  • Hubert Chan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3221)

Abstract

Upward planarity testing, or checking whether a directed graph has a drawing in which no edges cross and all edges point upward, is NP-complete. All of the algorithms for upward planarity testing developed previously focused on special classes of graphs. In this paper we develop a parameterized algorithm for upward planarity testing that can be applied to all graphs and runs in O(f(k)n 3 + g(k,ℓ)n) time, where n is the number of vertices, k is the number of triconnected components, and ℓ is the number of cutvertices. The functions f(k) and g(k,ℓ) are defined as f(k)=k!8 k and \(g(k,\ell)=2^{3\cdot 2^\ell}k^{3\cdot 2^\ell} k!8^k\). Thus if the number of triconnected components and the number of cutvertices are small, the problem can be solved relatively quickly, even for a large number of vertices. This is the first parameterized algorithm for upward planarity testing.

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References

  1. 1.
    Bertolazzi, P., Battista, G.D., Didimo, W.: Quasi-upward planarity. Algorithmica 32, 474–506 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bertolazzi, P., Battista, G.D., Liotta, G., Mannino, C.: Upward drawings of triconnected digraphs. Algorithmica 12, 476–497 (1994)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Bertolazzi, P., Battista, G.D., Mannino, C., Tamassia, R.: Optimal upward planarity testing of single-source digraphs. SIAM Journal on Computing 27(1), 132–196 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bondy, J.A., Murty, U.S.R.: Graph Theory with Applications. North Holland, New York (1976)Google Scholar
  5. 5.
    Chan, H.: A parameterized algorithm for upward planarity testing of biconnected graphs. Master’s thesis, School of Computer Science, University of Waterloo (May 2003)Google Scholar
  6. 6.
    Battista, G.D., Eades, P., Tamassia, R., Tollis, I.G.: Graph Drawing: Algorithms for the Visualization of Graphs. Prentice-Hall, New Jersey (1999)zbMATHGoogle Scholar
  7. 7.
    Di Battista, G., Liotta, G.: Upward planarity checking: “Faces are more than polygons” (extended abstract). In: Whitesides, S.H. (ed.) GD 1998. LNCS, vol. 1547, pp. 72–86. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  8. 8.
    Di Battista, G., Liu, W.-P., Rival, I.: Bipartite graphs, upward drawings, and planarity. Information Processing Letters 36, 317–322 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Di Battista, G., Tamassia, R.: Algorithms for plane representations of acyclic digraphs. Theoretical Computer Science 61, 175–198 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Di Battista, G., Tamassia, R., Tollis, I.G.: Area requirement and symmetry display of planar upward drawings. Discrete and Computational Geometry 7, 381–401 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Diestel, R.: Graph Theory, 2nd edn. Graduate Texts in Mathematics, vol. 173. Springer, New York (2000)Google Scholar
  12. 12.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Monographs in Computer Science. Springer, New York (1997)Google Scholar
  13. 13.
    Dujmović, V., Fellows, M.R., Hallett, M.T., Kitching, M., Liotta, G., McCartin, C., Nishimura, N., Ragde, P., Rosamond, F., Suderman, M., Whitesides, S., Wood, D.R.: A fixed-parameter approach to two-layer planarization. In: Mutzel, P., Jünger, M., Leipert, S. (eds.) GD 2001. LNCS, vol. 2265, pp. 1–15. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  14. 14.
    Dujmović, V., Fellows, M.R., Hallett, M.T., Kitching, M., Liotta, G., McCartin, C., Nishimura, N., Ragde, P., Rosamond, F., Suderman, M., Whitesides, S., Wood, D.R.: On the parameterized complexity of layered graph drawing. In: Proceedings, 9th Annual European Symposium on Algorithms, pp. 488–499 (2001)Google Scholar
  15. 15.
    Dujmović, V., Fernau, H., Kaufmann, M.: Fixed parameter algorithms for one-sided crossing minimization revisited. In: Liotta, G. (ed.) GD 2003. LNCS, vol. 2912, pp. 332–344. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  16. 16.
    Dujmović, V., Morin, P., Wood, D.R.: Path-width and three-dimensional straight-line grid drawings of graphs. In: Goodrich, M.T., Kobourov, S.G. (eds.) GD 2002. LNCS, vol. 2528, pp. 42–53. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  17. 17.
    Garg, A., Tamassia, R.: On the computational complexity of upward and rectilinear planarity testing. SIAM Journal on Computing 31(2), 601–625 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Hopcroft, J., Tarjan, R.E.: Dividing a graph into triconnected compononts. SIAM Journal on Computing 2, 136–158 (1972)MathSciNetGoogle Scholar
  19. 19.
    Hutton, M.D., Lubiw, A.: Upward planar drawing of single source acyclic digraphs. SIAM Journal on Computing 25(2), 291–311 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Jünger, M., Leipert, S., Mutzel, P.: Level planarity testing in linear time. In: Whitesides, S.H. (ed.) GD 1998. LNCS, vol. 1547, pp. 224–237. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  21. 21.
    Kaufmann, M., Wagner, D. (eds.): Drawing Graphs: Methods and Models. LNCS, vol. 2025. Springer, Heidelberg (2001)zbMATHGoogle Scholar
  22. 22.
    Papakostas, A.: Upward planarity testing of outerplanar dags (extended abstract). In: Tamassia, R., Tollis, I.G. (eds.) GD 1994. LNCS, vol. 894, pp. 298–306. Springer, Heidelberg (1995)Google Scholar
  23. 23.
    Peng, Z.: Drawing graphs of bounded treewidth/pathwidth. Master’s thesis, Department of Computer Science, University of Auckland (2001)Google Scholar
  24. 24.
    Suderman, M., Whitesides, S.: Experiments with the fixed-parameter approach for two-layer planarization. In: Liotta, G. (ed.) GD 2003. LNCS, vol. 2912, pp. 345–356. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  25. 25.
    Tamassia, R., Tollis, I.G.: A unified approach to visibility representations of planar graphs. Discrete and Computational Geometry 1(4), 321–341 (1986)zbMATHMathSciNetGoogle Scholar
  26. 26.
    Tarjan, R.E.: Depth-first searcth and linear graph algorithms. SIAM Journal on Computing 1(2), 145–159 (1972)CrossRefMathSciNetGoogle Scholar
  27. 27.
    Thomassen, C.: Planar acyclic oriented graphs. Order 5, 349–361 (1989)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Hubert Chan
    • 1
  1. 1.School of Computer ScienceUniversity of WaterlooCanada

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