Canonical Ordering for Triangulations on the Cylinder, with Applications to Periodic Straight-Line Drawings

  • Luca Castelli Aleardi
  • Olivier Devillers
  • Éric Fusy
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7704)


We extend the notion of canonical orderings to cylindric triangulations. This allows us to extend the incremental straight-line drawing algorithm of de Fraysseix, Pach and Pollack to this setting. Our algorithm yields in linear time a crossing-free straight-line drawing of a cylindric triangulation G with n vertices on a regular grid ℤ/wℤ×[0..h], with w ≤ 2n and h ≤ n(2d + 1), where d is the (graph-) distance between the two boundaries. As a by-product, we can also obtain in linear time a crossing-free straight-line drawing of a toroidal triangulation with n vertices on a periodic regular grid ℤ/wℤ×ℤ/hℤ, with w ≤ 2n and h ≤ 1 + n(2c + 1), where c is the length of a shortest non-contractible cycle. Since \(c\leq\sqrt{2n}\), the grid area is O(n 5/2). Our algorithms apply to any triangulation (whether on the cylinder or on the torus) that have no loops nor multiple edges in the periodic representation.


  1. 1.
    Albertson, M.O., Hutchinson, J.P.: On the independence ratio of a graph. J. Graph. Theory 2, 1–8 (1978)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bonichon, N., Gavoille, C., Labourel, A.: Edge partition of toroidal graphs into forests in linear time. In: ICGT, vol. 22, pp. 421–425 (2005)Google Scholar
  3. 3.
    Brehm, E.: 3-orientations and Schnyder 3-trees decompositions, Master’s thesis, FUB (2000)Google Scholar
  4. 4.
    Castelli-Aleardi, L., Fusy, E., Lewiner, T.: Schnyder woods for higher genus triangulated surfaces, with applications to encoding. Discr. & Comp. Geom. 42(3), 489–516 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Chambers, E., Eppstein, D., Goodrich, M., Loffler, M.: Drawing graphs in the plane with a prescribed outer face and polynomial area. JGAA 16(2), 243–259 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    de Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10(1), 41–51 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Duncan, C., Goodrich, M., Kobourov, S.: Planar drawings of higher-genus graphs. Journal of Graph Algorithms and Applications 15, 13–32 (2011)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Gonçalves, D., Lévêque, B.: Toroidal maps: Schnyder woods, orthogonal surfaces and straight-line representations arXiv:1202.0911 (2012)Google Scholar
  9. 9.
    Gortler, S.J., Gotsman, C., Thurston, D.: Discrete one-forms on meshes and applications to 3D mesh parameterization. Computer Aided Geometric Design 23(2), 83–112 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Kant, G.: Drawing planar graphs using the canonical ordering. Algorithmica 16(1), 4–32 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Kocay, W., Neilson, D., Szypowski, R.: Drawing graphs on the torus. Ars Combinatoria 59, 259–277 (2001)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Mohar, B.: Straight-line representations of maps on the torus and other flat surfaces. Discrete Mathematics 15, 173–181 (1996)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Mohar, B., Rosenstiehl, P.: Tessellation and visibility representations of maps on the torus. Discrete & Comput. Geom. 19, 249–263 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Schnyder, W.: Embedding planar graphs on the grid. In: SODA, pp. 138–148 (1990)Google Scholar
  15. 15.
    Zitnik, A.: Drawing graphs on surfaces. SIAM J. Disc. Math. 7(4), 593–597 (1994)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Luca Castelli Aleardi
    • 1
  • Olivier Devillers
    • 2
  • Éric Fusy
    • 1
  1. 1.LIX - École PolytechniqueFrance
  2. 2.INRIA Sophia AntipolisMéditerranéeFrance

Personalised recommendations