Advertisement

On Simultaneous Planar Graph Embeddings

  • P. Brass
  • E. Cenek
  • C. A. Duncan
  • A. Efrat
  • C. Erten
  • D. Ismailescu
  • S. G. Kobourov
  • A. Lubiw
  • J. S. B. Mitchell
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2748)

Abstract

We consider the problem of simultaneous embedding of planar graphs. There are two variants of this problem, one in which the mapping between the vertices of the two graphs is given and another in which the mapping is not given. In particular, given a mapping, we show how to embed two paths on an n ×n grid, and two caterpillar graphs on a 3n ×3n grid. We show that it is not always possible to simultaneously embed three paths. If the mapping is not given, we show that any number of outerplanar graphs can be embedded simultaneously on an O(n) ×O(n) grid, and an outerplanar and general planar graph can be embedded simultaneously on an O(n 2) ×O(n 2) grid.

Keywords

Planar Graph Outerplanar Graph Convex Drawing Grid Drawing Related Graph Property 
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.
    Bern, M., Gilbert, J.R.: Drawing the planar dual. Information Processing Letters 43(1), 7–13 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bernhart, F., Kainen, P.C.: The book thickness of a graph. J. Combin. Theory, Ser. B 27, 320–331 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bose, P.: On embedding an outer-planar graph in a point set. CGTA: Computational Geometry: Theory and Applications 23(3), 303–312 (2002)zbMATHGoogle Scholar
  4. 4.
    Brightwell, G.R., Scheinerman, E.R.: Representations of planar graphs. SIAM Journal on Discrete Mathematics 6(2), 214–229 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Cenek, E.: Layered and Stratified Graphs. PhD thesis, University of Waterloo (forthcoming) Google Scholar
  6. 6.
    Chrobak, M., Goodrich, M.T., Tamassia, R.: Convex drawings of graphs in two and three dimensions. In: Proc. 12th Annu. ACM Sympos. Comput. Geom., pp. 319–328 (1996)Google Scholar
  7. 7.
    Chrobak, M., Kant, G.: Convex grid drawings of 3-connected planar graphs. Intl. Journal of Computational Geometry and Applications 7(3), 211–223 (1997)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Collberg, C., Kobourov, S.G., Nagra, J., Pitts, J., Wampler, K.: A system for graph-based visualization of the evolution of software. In: 1st ACM Symposium on Software Visualization (2003) (to appear)Google Scholar
  9. 9.
    de Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10(1), 41–51 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Di Battista, G., Eades, P., Tamassia, R., Tollis, I.G.: Graph Drawing: Algorithms for the Visualization of Graphs. Prentice Hall, Englewood Cliffs (1999)zbMATHGoogle Scholar
  11. 11.
    Dillencourt, M.B., Eppstein, D., Hirschberg, D.S.: Geometric thickness of complete graphs. Journal of Graph Algorithms and Applications 4(3), 5–17 (2000)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Erdös, P.: Appendix. In: Roth, K.F. (ed.) On a problem of Heilbronn. J. London Math. Soc., vol. 26, pp. 198–204 (1951)Google Scholar
  13. 13.
    Erten, C., Kobourov, S.G.: Simultaneous embedding of a planar graph and its dual on the grid. In: 13th Intl. Symp. on Algorithms and Computation (ISAAC), pp. 575–587 (2002)Google Scholar
  14. 14.
    Gritzmann, P., Mohar, B., Pach, J., Pollack, R.: Embedding a planar triangulation with vertices at specified points. American Math. Monthly 98, 165–166 (1991)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Kaufmann, M., Wagner, D.: Drawing Graphs. LNCS, vol. 2025. Springer, New York (2001)zbMATHCrossRefGoogle Scholar
  16. 16.
    Koebe, P.: Kontaktprobleme der konformen Abbildung. Berichte uber die Verhandlungen der Sächsischen Akademie der Wissenschaften zu Leipzig. Math. Phys. Klasse 88, 141–164 (1936)Google Scholar
  17. 17.
    Miura, K., Nakano, S.-I., Nishizeki, T.: Grid drawings of 4-connected plane graphs. Discrete and Computational Geometry 26(1), 73–87 (2001)zbMATHMathSciNetGoogle Scholar
  18. 18.
    Mutzel, P., Odenthal, T., Scharbrodt, M.: The thickness of graphs: a survey. Graphs Combin. 14(1), 59–73 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Schnyder, W.: Planar graphs and poset dimension. Order 5(4), 323–343 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Tutte, W.T.: How to draw a graph. Proc. London Math. Society 13(52), 743–768 (1963)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Yannakakis, M.: Embedding planar graphs in four pages. Journal of Computer and System Sciences 38(1), 36–67 (1989)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • P. Brass
    • 1
  • E. Cenek
    • 2
  • C. A. Duncan
    • 3
  • A. Efrat
    • 4
  • C. Erten
    • 4
  • D. Ismailescu
    • 5
  • S. G. Kobourov
    • 4
  • A. Lubiw
    • 2
  • J. S. B. Mitchell
    • 6
  1. 1.Dept. of Computer ScienceCity College of New York 
  2. 2.Dept. of Computer ScienceUniversity of Waterloo 
  3. 3.Dept. of Computer ScienceUniv. of Miami 
  4. 4.Dept. of Computer ScienceUniv. of Arizona 
  5. 5.Dept. of MathematicsHofstra University 
  6. 6.Dept. of Applied Mathematics and StatisticsStony Brook University 

Personalised recommendations