Convex Drawings of 3-Connected Plane Graphs

  • Nicolas Bonichon
  • Stefan Felsner
  • Mohamed Mosbah
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3383)


We use Schnyder woods of 3-connected planar graphs to produce convex straight line drawings on a grid of size (n − 2 − Δ) ×(n − 2 − Δ). The parameter Δ≥ 0 depends on the the Schnyder wood used for the drawing. This parameter is in the range \(0 \leq \Delta\leq \frac{n}{2}-2\).


  1. 1.
    Rote, G.: Strictly convex drawings of planar graphs (2004) Google Scholar
  2. 2.
    Rosenstiehl, P., Tarjan, R.E.: Rectilinear planar layouts and bipolar orientations of planar graphs. Discrete Comput. Geom. 1, 343–353 (1986)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Schnyder, W.: Planar graphs and poset dimension. Order 5, 323–343 (1989)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    de Fraysseix, H., Pach, J., Pollack, R.: Small sets supporting Fary embeddings of planar graphs. In: Proc. 20th Annu. ACM Sympos. Theory Comput., pp. 426–433 (1988)Google Scholar
  5. 5.
    de Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10, 41–51 (1990)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    He, X.: Grid embeddings of 4-connected plane graphs. Discrete Comput. Geom. 17, 339–358 (1997)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Miura, K., Nakano, S., Nishizeki, T.: Grid drawings of 4-connected plane graphs. Discrete Comput. Geom. 26, 73–87 (2001)MATHMathSciNetGoogle Scholar
  8. 8.
    Schnyder, W.: Embedding planar graphs on the grid. In: Proc. 1st ACM-SIAM Sympos. Discrete Algorithms, pp. 138–148 (1990)Google Scholar
  9. 9.
    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
  10. 10.
    Tutte, W.T.: Convex representations of graphs. Proceedings London Mathematical Society 10, 304–320 (1960)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Tutte, W.T.: How to draw a graph. Proceedings London Mathematical Society 13, 743–768 (1963)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Kant, G.: Drawing planar graphs using the canonical ordering. Algorithmica 16, 4–32 (1996)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Chrobak, M., Kant, G.: Convex grid drawings of 3-connected planar graphs. Internat. J. Comput. Geom. Appl. 7, 211–223 (1997)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Schnyder, W., Trotter, W.T.: Convex embeddings of 3-connected plane graphs. Abstracts of the AMS 13, 502 (1992)Google Scholar
  15. 15.
    Di Battista, G., Tamassia, R., Vismara, L.: Output-sensitive reporting of disjoint paths. Algorithmica 23, 302–340 (1999)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Felsner, S.: Convex drawings of planar graphs and the order dimension of 3-polytopes. Order, 19–37 (2001)Google Scholar
  17. 17.
    Felsner, S.: Geometric Graphs and Arrangements. Vieweg Verlag (2004) Google Scholar
  18. 18.
    Felsner, S.: Lattice structures from planar graphs. Electron. J. Comb. 11 R15, 24p. (2004)MathSciNetGoogle Scholar
  19. 19.
    Fusy, E., Poulalhon, D., Schaeffer, G.: Coding, counting and sampling 3-connected planar graphs. In: 16th ACM-SIAM Sympos. Discrete Algorithms (2005) (to appear)Google Scholar
  20. 20.
    Bonichon, N., Gavoille, C., Hanusse, N., Poulalhon, D., Schaeffer, G.: Planar graphs, via well-orderly maps and trees. In: Hromkovič, J., Nagl, M., Westfechtel, B. (eds.) WG 2004. LNCS, vol. 3353, pp. 270–284. Springer, Heidelberg (2004)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Nicolas Bonichon
    • 1
  • Stefan Felsner
    • 2
  • Mohamed Mosbah
    • 1
  1. 1.LaBRI , Université Bordeaux-1Talence CedexFrance
  2. 2.Institut für Mathematik, MA 6-1Technische Universität BerlinBerlinGermany

Personalised recommendations