Triangle Contact Representations and Duality

  • Daniel Gonçalves
  • Benjamin Lévêque
  • Alexandre Pinlou
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6502)


A contact representation by triangles of a graph is a set of triangles in the plane such that two triangles intersect on at most one point, each triangle represents a vertex of the graph and two triangles intersects if and only if their corresponding vertices are adjacent. de Fraysseix, Ossona de Mendez and Rosenstiehl proved that every planar graph admits a contact representation by triangles. We strengthen this in terms of a simultaneous contact representation by triangles of a planar map and of its dual.

A primal-dual contact representation by triangles of a planar map is a contact representation by triangles of the primal and a contact representation by triangles of the dual such that for every edge uv, bordering faces f and g, the intersection between the triangles corresponding to u and v is the same point as the intersection between the triangles corresponding to f and g. We prove that every 3-connected planar map admits a primal-dual contact representation by triangles. Moreover, the interiors of the triangles form a tiling of the triangle corresponding to the outer face and each contact point is a node of exactly three triangles. Then we show that these representations are in one-to-one correspondence with generalized Schnyder woods defined by Felsner for 3-connected planar maps.


  1. 1.
    Andreev, E.: On convex polyhedra in Lobachevsky spaces. In: Mat. Sbornik, Ser., vol. 81, pp. 445–478 (1970)Google Scholar
  2. 2.
    Badent, M., Binucci, C., Di Giacomo, E., Didimo, W., Felsner, S., Giordano, F., KratochvÃI, J. Palladino, P., Patrignani, M., Trotta., F.: Homothetic triangle contact representations of planar graphs. In: Proceedings of the 19th Canadian Conference on Computational Geometry CCCG 2007, pp. 233–236 (2007)Google Scholar
  3. 3.
    Bonichon, N., Felsner, S., Mosbah, M.: Convex Drawings of 3-Connected Plane Graphs. Algorithmica 47, 399–420 (2007)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Felsner, S.: Convex Drawings of Planar Graphs and the Order Dimension of 3-Polytopes. Order 18, 19–37 (2001)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Felsner, S.: Geodesic Embeddings and Planar Graphs. Order 20, 135–150 (2003)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Felsner, S.: Lattice structures from planar graphs. Electron. J. Combin. 11 (2004)Google Scholar
  7. 7.
    Felsner, S., Zickfeld, F.: Schnyder Woods and Orthogonal Surfaces. Discrete Comput. Geom. 40, 103–126 (2008)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    de Fraysseix, H., Ossona de Mendez, P., Rosenstiehl, P.: On Triangle Contact Graphs. Combinatorics, Probability and Computing 3, 233–246 (1994)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    de Fraysseix, H., Ossona de Mendez, P.: On topological aspects of orientations. Discrete Mathematics 229, 57–72 (2001)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    de Fraysseix, H., de Mendez, P.O.: Barycentric systems and stretchability. Discrete Applied Mathematics 155, 1079–1095 (2007)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Gansner, E.R., Hu, Y., Kaufmann, M., Kobourov, S.G.: Optimal Polygonal Representation of Planar Graphs. In: López-Ortiz, A. (ed.) LATIN 2010. LNCS, vol. 6034, pp. 417–432. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  12. 12.
    Gansner, E.R., Hu, Y., Kobourov, S.G.: On Touching Triangle Graphs. In: Proc. Graph Drawing (2010),
  13. 13.
    Kaufmann, M., Kratochvíl, J., Lehmann, K.A., Subramanian, A.R.: Max-tolerance graphs as intersection graphs: Cliques, cycles and recognition. In: Proc. SODA 2006, pp. 832–841 (2006)Google Scholar
  14. 14.
    Koebe, P.: Kontaktprobleme der konformen Abbildung. Berichte Äuber die Verhandlungen d. SÄachs. Akad. d. Wiss., Math.-Phys. Klasse 88, 141–164 (1936)MATHGoogle Scholar
  15. 15.
    Kratochvíl, J.: Bertinoro Workshop on Graph Drawing (2007)Google Scholar
  16. 16.
    Miller, E.: Planar graphs as minimal resolutions of trivariate monomial ideals. Documenta Mathematica 7, 43–90 (2002)MathSciNetMATHGoogle Scholar
  17. 17.
    Schnyder, W.: Planar graphs and poset dimension. Order 5, 323–343 (1989)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Schramm, O.: Combinatorically Prescribed Packings and Applications to Conformal and Quasiconformal Maps, Modified version of PhD thesis from (1990),

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Daniel Gonçalves
    • 1
  • Benjamin Lévêque
    • 1
  • Alexandre Pinlou
    • 1
  1. 1.LIRMMCNRS & Univ. Montpellier 2Montpellier

Personalised recommendations