Skip to main content
SpringerLink
Log in

On Simultaneous Planar Graph Embeddings

  • Conference paper
Algorithms and Data Structures (WADS 2003)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2748))

Included in the following conference series:

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bern, M., Gilbert, J.R.: Drawing the planar dual. Information Processing Letters 43(1), 7–13 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bernhart, F., Kainen, P.C.: The book thickness of a graph. J. Combin. Theory, Ser. B 27, 320–331 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bose, P.: On embedding an outer-planar graph in a point set. CGTA: Computational Geometry: Theory and Applications 23(3), 303–312 (2002)

    MATH  Google Scholar 

  4. Brightwell, G.R., Scheinerman, E.R.: Representations of planar graphs. SIAM Journal on Discrete Mathematics 6(2), 214–229 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  5. Cenek, E.: Layered and Stratified Graphs. PhD thesis, University of Waterloo (forthcoming)

    Google Scholar 

  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. Chrobak, M., Kant, G.: Convex grid drawings of 3-connected planar graphs. Intl. Journal of Computational Geometry and Applications 7(3), 211–223 (1997)

    Article  MathSciNet  Google Scholar 

  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. de Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10(1), 41–51 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  10. Di Battista, G., Eades, P., Tamassia, R., Tollis, I.G.: Graph Drawing: Algorithms for the Visualization of Graphs. Prentice Hall, Englewood Cliffs (1999)

    MATH  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  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. 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. Gritzmann, P., Mohar, B., Pach, J., Pollack, R.: Embedding a planar triangulation with vertices at specified points. American Math. Monthly 98, 165–166 (1991)

    Article  MathSciNet  Google Scholar 

  15. Kaufmann, M., Wagner, D.: Drawing Graphs. LNCS, vol. 2025. Springer, New York (2001)

    Book  MATH  Google Scholar 

  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. Miura, K., Nakano, S.-I., Nishizeki, T.: Grid drawings of 4-connected plane graphs. Discrete and Computational Geometry 26(1), 73–87 (2001)

    MATH  MathSciNet  Google Scholar 

  18. Mutzel, P., Odenthal, T., Scharbrodt, M.: The thickness of graphs: a survey. Graphs Combin. 14(1), 59–73 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  19. Schnyder, W.: Planar graphs and poset dimension. Order 5(4), 323–343 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  20. Tutte, W.T.: How to draw a graph. Proc. London Math. Society 13(52), 743–768 (1963)

    Article  MATH  MathSciNet  Google Scholar 

  21. Yannakakis, M.: Embedding planar graphs in four pages. Journal of Computer and System Sciences 38(1), 36–67 (1989)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dept. of Computer Science, City College of New York,  

    P. Brass

  2. Dept. of Computer Science, University of Waterloo,  

    E. Cenek & A. Lubiw

  3. Dept. of Computer Science, Univ. of Miami,  

    C. A. Duncan

  4. Dept. of Computer Science, Univ. of Arizona,  

    A. Efrat, C. Erten & S. G. Kobourov

  5. Dept. of Mathematics, Hofstra University,  

    D. Ismailescu

  6. Dept. of Applied Mathematics and Statistics, Stony Brook University,  

    J. S. B. Mitchell

Authors
  1. P. Brass
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. Cenek
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. C. A. Duncan
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. A. Efrat
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. C. Erten
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. D. Ismailescu
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. S. G. Kobourov
    View author publications

    You can also search for this author in PubMed Google Scholar

  8. A. Lubiw
    View author publications

    You can also search for this author in PubMed Google Scholar

  9. J. S. B. Mitchell
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Carleton University, Ottawa, Canada

    Frank Dehne

  2. School of Computer Science, Carleton University, 1125 Colonel By Drive, K1S 5B6, Ottawa, Canada

    Jörg-Rüdiger Sack

  3. School of Computer Science, Carleton University, K1S 5B6, Ottawa, ON, Canada

    Michiel Smid

Rights and permissions

Reprints and permissions

Copyright information

© 2003 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Brass, P. et al. (2003). On Simultaneous Planar Graph Embeddings. In: Dehne, F., Sack, JR., Smid, M. (eds) Algorithms and Data Structures. WADS 2003. Lecture Notes in Computer Science, vol 2748. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-45078-8_22

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-45078-8_22

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-40545-0

  • Online ISBN: 978-3-540-45078-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation