An Application of Well-Orderly Trees in Graph Drawing

  • Huaming Zhang
  • Xin He
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3843)


Well-orderly trees seem to have the potential of becoming a powerful technique capable of deriving new results in graph encoding, graph enumeration and graph generation [3, 4]. In this paper, we reduce the height of the visibility representation of plane graphs from 5n/6 to (4n–1)/5, by using well-orderly trees.


  1. 1.
    di Battista, G., Eades, P., Tammassia, R., Tollis, I.: Graph Drawing: Algorithms for the Visualization of Graphs. Prentice Hall, Englewood Cliffs (1998)Google Scholar
  2. 2.
    Bonichon, N., Le Saëc, B., Mosbah, M.: Wagner’s Theorem on Realizers. In: Widmayer, P., Triguero, F., Morales, R., Hennessy, M., Eidenbenz, S., Conejo, R. (eds.) ICALP 2002. LNCS, vol. 2380, pp. 1043–1053. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  3. 3.
    Bonichon, N., Gavoille, C., Hanusse, N.: An information-theoretic upper bound of planar graphs using triangulation. In: Alt, H., Habib, M. (eds.) STACS 2003. LNCS, vol. 2607, pp. 499–510. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  4. 4.
    Bonichon, N., Gavoille, C., Hanusse, N.: Canonical decomposition of outerplanar maps and application to enumeration, coding and generation. In: Bodlaender, H.L. (ed.) WG 2003. LNCS, vol. 2880, pp. 81–92. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  5. 5.
    Chen, H.-L., Liao, C.-C., Lu, H.-I., Yen, H.-C.: Some applications of orderly spanning trees in graph drawing. In: Goodrich, M.T., Kobourov, S.G. (eds.) GD 2002. LNCS, vol. 2528, pp. 332–343. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  6. 6.
    Chiang, Y.-T., Lin, C.-C., Lu, H.-I.: Orderly spanning trees with applications to graph encoding and graph drawing. In: Proc. of the 12th Annual ACM-SIAM SODA, pp. 506–515. ACM Press, New York (2001)Google Scholar
  7. 7.
    Chrobak, M., Kant, G.: Convex grid drawings of 3-connected planar graphs, Technical Report RUU-CS-93-45, Department of Computer Science, Utrecht University, Holland (1993)Google Scholar
  8. 8.
    Fößmeier, U., Kant, G., Kaufmann, M.: 2-Visibility drawings of planar graphs. In: North, S.C. (ed.) GD 1996. LNCS, vol. 1190, pp. 155–168. Springer, Heidelberg (1997)Google Scholar
  9. 9.
    de Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10, 41–51 (1990)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Kant, G.: Drawing planar graphs using the lmc-ordering. In: Proc. 33rd Symposium on Foundations of Computer Science, pp. 101–110. IEEE, Pittsburgh (1992)CrossRefGoogle Scholar
  11. 11.
    Kant, G.: Algorithms for drawing planar graphs, Ph.D. Dissertation, Department of Computer Science, University of Utrecht, Holland (1993)Google Scholar
  12. 12.
    Kant, G.: Drawing planar graphs using the canonical ordering. Algorithmica 16, 4–32 (1996)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Kant, G.: A more compact visibility representation. International Journal of Computational Geometry and Applications 7, 197–210 (1997)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Kant, G., He, X.: Regular edge labeling of 4-connected plane graphs and its applications in graph drawing problems. Theoretical Computer Science 172, 175–193 (1997)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Lempel, A., Even, S., Cederbaum, I.: An algorithm for planarity testing of graphs. In: Theory of Graphs (Proc. of an International Symposium), Rome, July 1966, pp. 215–232 (1967)Google Scholar
  16. 16.
    Liao, C.-C., Lu, H.-I., Yen, H.-C.: Floor-planning via orderly spanning trees. In: Mutzel, P., Jünger, M., Leipert, S. (eds.) GD 2001. LNCS, vol. 2265, pp. 367–377. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  17. 17.
    Lin, C.-C., Lu, H.-I., Sun, I.-F.: Improved compact visibility representation of planar graph via Schnyder’s realizer. SIAM Journal on Discrete Mathematics 18, 19–29 (2004)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Ossona de Mendez, P.: Orientations bipolaires. PhD thesis, Ecole des Hautes Etudes en Sciences Sociales, Paris (1994)Google Scholar
  19. 19.
    Rosenstiehl, P., Tarjan, R.E.: Rectilinear planar layouts and bipolar orientations of planar graphs. Discrete Comput. Geom. 1, 343–353 (1986)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Schnyder, W.: Planar graphs and poset dimension. Order 5, 323–343 (1989)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Schnyder, W.: Embedding planar graphs on the grid. In: Proc. of the First Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 138–148. SIAM, Philadelphia (1990)Google Scholar
  22. 22.
    Tamassia, R., Tollis, I.G.: An unified approach to visibility representations of planar graphs. Discrete Comput. Geom. 1, 321–341 (1986)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Zhang, H., He, X.: Compact Visibility Representation and Straight-Line Grid Embedding of Plane Graphs. In: Dehne, F., Sack, J.-R., Smid, M. (eds.) WADS 2003. LNCS, vol. 2748, pp. 493–504. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  24. 24.
    Zhang, H., He, X.: On Visibility Representation of Plane Graphs. In: Diekert, V., Habib, M. (eds.) STACS 2004. LNCS, vol. 2996, pp. 477–488. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  25. 25.
    Zhang, H., He, X.: New Theoretical Bounds of Visibility Representation of Plane Graphs. In: Pach, J. (ed.) GD 2004. LNCS, vol. 3383, pp. 425–430. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  26. 26.
    Zhang, H., He, X.: Improved Visibility Representation of Plane Graphs. Discrete Comput. Geom. 30, 29–29 (2005)MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Huaming Zhang
    • 1
  • Xin He
    • 2
  1. 1.Department of Computer ScienceUniversity of Alabama in HuntsvilleHuntsvilleUSA
  2. 2.Department of Computer Science and EngineeringSUNY at BuffaloBuffaloUSA

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