Order

, Volume 12, Issue 2, pp 109–133 | Cite as

Upward planarity testing

  • Ashim Garg
  • Roberto Tamassia
Article

Abstract

Acyclic digraphs, such as the covering digraphs of ordered sets, are usually drawn upward, i.e., with the edges monotonically increasing in the vertical direction. A digraph is upward planar if it admits an upward planar drawing. In this survey paper, we overview the literature on the problem of upward planarity testing. We present several characterizations of upward planarity and describe upward planarity testing algorithms for special classes of digraphs, such as embedded digraphs and single-source digraphs. We also sketch the proof of NP-completeness of upward planarity testing.

Mathemtics Subject Classifications (1991)

05C10 06A99 68Q20 68R10 68U05 

Key words

Graph drawing ordered set lattice upward planarity NP-completeness directed acyclic graph 

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References

  1. 1.
    Baker, K. A., Fishburn, P., and Roberts, F. S. (1971) Partial orders of dimension 2,Networks 2, 11–28.Google Scholar
  2. 2.
    Bertolazzi, P., Cohen, R. F., Di Battista, G., Tamassia, R., and Tollis, I. G. (1994) How to draw a series-parallel digraph,Internat. J. Comput. Geom. Appl. 4, 385–402.Google Scholar
  3. 3.
    Bertolazzi, P. and Di Battista, G. (1991) On upward drawing testing of triconnected digraphs, inProc. 7th Annu. ACM Sympos. Comput. Geom, pp. 272–280.Google Scholar
  4. 4.
    Bertolazzi, P., Di Battista, G, Liotta, G., and Mannino, C. (1994) Upward drawings of triconnected digraphs,Algorithmica 12, 476–497.Google Scholar
  5. 5.
    Bertolazzi, P., Di Battista, G., Mannino, C., and Tamassia, R. (1993) Optimal upward planarity testing of single-source digraphs, in1st Annual European Symposium on Algorithms (ESA '93), Vol. 726 ofLecture Notes in Computer Science, Springer-Verlag, pp. 37–48.Google Scholar
  6. 6.
    Birkhoff, G. (1967)Lattice Theory, American Mathematical Society, Providence, RI.Google Scholar
  7. 7.
    Bondy, J. A. and Murty, U. S. R. (1976)Graph Theory with Applications, North-Holland, New York, NY.Google Scholar
  8. 8.
    Booth, K. and Lueker, G. (1976) Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms,J. Comput. Syst. Sci. 13, 335–379.Google Scholar
  9. 9.
    Cormen, T. H., Leiserson, C. E., and Rivest, R. L. (1990)Introduction to Algorithms, The MIT Press, Cambridge, Mass.Google Scholar
  10. 10.
    de Fraysseix, H. and Rosenstiehl, P. (1982) A depth-first-search characterization of planarity,Annals of Dicrete Mathematics 13, 75–80.Google Scholar
  11. 11.
    Di Battista, G., Eades, P., Tamassia, R., and Tollis, I. G. (1994) Algorithms for drawing graphs: an annotated bibliography,Comput. Geom. Theory Appl. 4, 235–282.Google Scholar
  12. 12.
    Di Battista, G., Liu, W. P., and Rival, I. (1990) Bipartite graphs, upward drawings, and planarity,Inform. Process. Lett. 36, 317–322.Google Scholar
  13. 13.
    Di Battista, G. and Tamassia, R. (1988) Algorithms for plane representations of acyclic digraphs,Theoret. Comput. Sci. 61, 175–198.Google Scholar
  14. 14.
    Di Battista, G. and Tamassia, R. (1990) On-line graph algorithms with SPQR-trees, inAutomata, Languages and Programming (Proc. 17th ICALP), Vol. 442 ofLecture Notes in Computer Science, pp. 598–611.Google Scholar
  15. 15.
    Di Battista, G., Tamassia, R., and Tollis, I. G. (1992) Area requirement and symmetry display of planar upward drawings,Discrete Comput. Geom. 7, 381–401.Google Scholar
  16. 16.
    Garey, M. R. and Johnson, D. S. (1979)Computers and Intractability: A Guide to the Theory of NP-Completeness, W. H. Freeman, New York, NY.Google Scholar
  17. 17.
    Garg, A. and Tamassia, R. (1995) On the computational complexity of upward and rectilinear planarity testing, in R. Tamassia and I. G. Tollis (eds),Graph Drawing (Proc. GD '94), Vol. 894 ofLecture Notes in Computer Science, Springer-Verlag, pp. 286–297.Google Scholar
  18. 18.
    Hopcroft, J. and Tarjan, R. E. (1973) Dividing a graph into triconnected components,SIAM J. Comput. 2, 135–158.Google Scholar
  19. 19.
    Hopcroft, J. and Tarjan, R. E. (1974) Efficient planarity testing,J. ACM 21(4), 549–568.Google Scholar
  20. 20.
    Hutton, M. D. and Lubiw, A. (1991) Upward planar drawing of single source acyclic digraphs, inProc. 2nd ACM-SIAM Sympos. Discrete Algorithms, pp. 203–211.Google Scholar
  21. 21.
    Kelly, D. (1987) Fundamentals of planar ordered sets,Discrete Math. 63, 197–216.Google Scholar
  22. 22.
    Kelly, D. and Rival, I. (1975) Planar lattices,Canad. J. Math. 27(3), 636–665.Google Scholar
  23. 23.
    Lempel, A., Even, S., and Cederbaum, I. (1967) An algorithm for planarity testing of graphs, inTheory of Graphs: Internat. Symposium (Rome 1966), Gordon and Breach, New York, pp. 215–232.Google Scholar
  24. 24.
    Papakostas, A. (1995) Upward planarity testing of outerplanar dags, in R. Tamassia and I. G. Tollis (eds),Graph Drawing (Proc. GD '94), Vol. 894 ofLecture Notes in Computer Science, Springer-Verlag, pp. 298–306.Google Scholar
  25. 25.
    Platt, C. (1976) Planar lattices and planar graphs,J. Combin. Theory Ser. B 21, 30–39.Google Scholar
  26. 26.
    Rival, I. (1993) Reading, drawing, and order, in I. G. Rosenberg and G. Sabidussi (eds),Algebras and Orders, Kluwer Academic Publishers, pp. 359–404.Google Scholar
  27. 27.
    Thomassen, C. (1989) Planar acyclic oriented graphs,Order 5(4), 349–361.Google Scholar

Copyright information

© Kluwer Academic Publishers 1995

Authors and Affiliations

  • Ashim Garg
    • 1
  • Roberto Tamassia
    • 1
  1. 1.Department of Computer ScienceBrown UniversityProvidenceU.S.A.

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