Advertisement

Compact Visibility Representation and Straight-Line Grid Embedding of Plane Graphs

  • Huaming Zhang
  • Xin He
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2748)

Abstract

We study the properties of Schnyder’s realizers and canonical ordering trees of plane graphs. Based on these newly discovered properties, we obtain compact drawings of two styles for any plane graph G with n vertices. First we show that G has a visibility representation with height at most \(\lceil \frac{15n}{16} \rceil\). This improves the previous best bound of n-1. The drawing can be obtained in linear time. Second, we show that every plane graph G has a straight-line grid embedding on an (n − Δ0 − 1) × (n − Δ0 − 1) grid, where Δ0 is the number of cyclic faces of G with respect to its minimum realizer. This improves the previous best bound of (n-1) × (n-1). This embedding can also be found in O(n) time.

Keywords

Plane Graph Unique Path Interior Vertex Graph Drawing Interior Face 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bonichon, N., Salëc, B.L., De la Libération, 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
  2. 2.
    Brehm, E.: 3-Orientations and Schnyder 3-tree-decompositions. Diploma Thesis, FB mathematik und Informatik, Freie Universität Berlin (2000) Google Scholar
  3. 3.
    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
  4. 4.
    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 (2001)Google Scholar
  5. 5.
    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 (1996)Google Scholar
  6. 6.
    de Fraysseix, H., Pach, J., Pollack, R.: Small sets supporting Straight-Line embeddings of planar graphs. In: Proc. 20th Annual Symposium on Theory of Computing, pp. 426–433 (1988)Google Scholar
  7. 7.
    de Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10, 41–51 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    He, X.: Grid Embedding of 4-connected Plane Graphs. Discrete Comput. Geom. 17, 339–358 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    He, X.: On floor-plan of plane graphs. SIAM Journal on Computing 28(6), 2150–2167 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    He, X., Kao, M.-Y., Lu, H.-I.: Linear-time succinct encodings of planar graphs via canonical orderings. SIAM Journal Discrete Math 12(3), 317–325 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Kant, G.: Drawing planar graphs using the lmc-ordering. In: Proc. 33rd Symposium on Foundations of Computer Science, Pittsburgh, pp. 101–110 (1992)Google Scholar
  12. 12.
    Kant, G.: Algorithms for Drawing Planar Graphs, Ph.D. Dissertation, Department of Computer Science, University of Utrecht (1993) Google Scholar
  13. 13.
    G. Kant, A more compact visibility representation. International Journal of Computational Geometry & Applications 7(3), pp. 197–210, 1997 Google Scholar
  14. 14.
    A. Lempel, S. Even and I. Cederbaum, An algorithm for planarity testing of graphs. Theory of Graphs (Proc. of an International Symposium, Rome, July 1966), pp. 215–232, 1967. Google Scholar
  15. 15.
    C.-C. Liao, H.-I. Lu and H.-C. Yen, Floor-planning via orderly spanning trees. Proc. 9th International Symposium on Graph Drawing (GD 2001), LNCS 2265, pp. 367–377, 2002. Google Scholar
  16. 16.
    C.-C. Lin, H.-I. Lu and I-F. Sun, Improved Compact Visibility Representation of Planar Graph via Schnyder’s Realizer. Proc. 20th Annual Symposium on Theoretical Aspects of Computer Science, LNCS 2607, pp. 14–25, 2003. Google Scholar
  17. 17.
    K. Miura, S. Nakano and T. Nishizeki, Grid Drawings of 4-Connected Plane Graphs. Discrete Comput. Geom. 26, pp. 73–87, 2001. Google Scholar
  18. 18.
    P. Rosenstiehl and R. E. Tarjan, Rectilinear planar layouts and bipolar orientations of planar graphs. Discrete Comput. Geom. 1, pp. 343–353, 1986. Google Scholar
  19. 19.
    W. Schnyder, Planar graphs and poset dimension. Order 5, pp. 323–343, 1989. Google Scholar
  20. 20.
    W. Schnyder, Embedding planar graphs on the grid. Proc. of the First Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 138–148, 1990. Google Scholar
  21. 21.
    R. Tamassia and I.G.Tollis, An unified approach to visibility representations of planar graphs. Discrete & Computational Geometry 1(4), pp. 321–341, 1986Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Huaming Zhang
    • 1
  • Xin He
    • 1
  1. 1.Department of Computer Science and EngineeringSUNY at BuffaloBuffaloUSA

Personalised recommendations