Schnyder Woods and Orthogonal Surfaces

  • Stefan Felsner
  • Florian Zickfeld
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4372)


In this paper we study connections between Schnyder woods and orthogonal surfaces. Schnyder woods and the face counting approach have important applications in graph drawing and dimension theory. Orthogonal surfaces explain the connections between these seemingly unrelated notions. We use these connections for an intuitive proof of the Brightwell-Trotter Theorem which says that the face lattice of a 3-polytope minus one face has dimension three. Our proof yields a companion linear time algorithm for the construction of the three linear orders that realize the face lattice.

Coplanar orthogonal surfaces are in correspondance with a large class of convex straight line drawings of 3-connected planar graphs. We show that Schnyder’s face counting approach with weighted faces can be used to construct all coplanar orthogonal surfaces and hence the corresponding drawings. Appropriate weights are computable in linear time.


Planar Graph Face Lattice Linear Time Algorithm Special Edge Schnyder Wood 
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.


  1. 1.
    Bárány, I., Rote, G.: Strictly convex drawings of planar graphs (2005), arXiv:cs.CG/0507030Google Scholar
  2. 2.
    Bonichon, N., Felsner, S., Mosbah, M.: Convex drawings of 3-connected planar graphs. In: Pach, J. (ed.) GD 2004. LNCS, vol. 3383, pp. 60–70. Springer, Heidelberg (2005)Google Scholar
  3. 3.
    Brightwell, G., Trotter, W.T.: The order dimension of convex polytopes. SIAM J. Discrete Math. 6, 230–245 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Brightwell, G., Trotter, W.T.: The order dimension of planar maps. SIAM J. Discrete Math. 10, 515–528 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Di Battista, G., Tamassia, R., Vismara, L.: Output-sensitive reporting of disjoint paths. Algorithmica 23, 302–340 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
  7. 7.
    Felsner, S.: Convex drawings of planar graphs and the order dimension of 3-polytopes. Order 18, 19–37 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Felsner, S.: Geodesic embeddings and planar graphs. Order 20, 135–150 (2003), zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Felsner, S.: Geometric Graphs and Arrangements. Vieweg, Wiesbaden (2004)zbMATHGoogle Scholar
  10. 10.
    Felsner, S., Zickfeld, F.: Schnyder woods and orthogonal surfaces (2006),
  11. 11.
    Fusy, E., Poulalhon, D., Schaeffer, G.: Dissection and trees, with applications to optimal mesh encoding and random sampling. In: Proc. 16. ACM-SIAM Sympos. Discrete Algorithms, pp. 690–699. ACM Press, New York (2005)Google Scholar
  12. 12.
    Kant, G.: Drawing planar graphs using the lmc-ordering. In: Proc. 33rd IEEE Sympos. on Found. of Comp. Sci., pp. 101–110. IEEE Computer Society Press, Los Alamitos (1992)CrossRefGoogle Scholar
  13. 13.
    Lin, C., Lu, H., Sun, I-F.: Improved compact visibility representation of planar graphs via Schnyder’s realizer. SIAM J. Discrete Math. 18, 19–29 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Miller, E.: Planar graphs as minimal resolutions of trivariate monomial ideals. Documenta Math. 7, 43–90 (2002)zbMATHGoogle Scholar
  15. 15.
    Schnyder, W.: Planar graphs and poset dimension. Order 5, 323–343 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Schnyder, W.: Embedding planar graphs on the grid. In: Proc. 1st ACM-SIAM Sympos. Discrete Algorithms, pp. 138–148. ACM Press, New York (1990)Google Scholar
  17. 17.
    Trotter, W.T.: Combinatorics and Partially Ordered Sets: Dimension Theory. The Johns Hopkins University Press, Baltimore (1992)zbMATHGoogle Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Stefan Felsner
    • 1
  • Florian Zickfeld
    • 1
  1. 1.Technische Universität Berlin, Fachbereich Mathematik, Straße des 17. Juni 136, 10623 BerlinGermany

Personalised recommendations