Straight Line Triangle Representations

  • Nieke Aerts
  • Stefan Felsner
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8242)


A straight line triangle representation (SLTR) of a planar graph is a straight line drawing such that all the faces including the outer face have triangular shape. Such a drawing can be viewed as a tiling of a triangle using triangles with the input graph as skeletal structure. In this paper we present a characterization of graphs that have an SLTR that is based on flat angle assignments, i.e., selections of angles of the graph that have size π in the representation. We also provide a second characterization in terms of contact systems of pseudosegments. With the aid of discrete harmonic functions we show that contact systems of pseudosegments that respect certain conditions are stretchable. The stretching procedure is then used to get straight line triangle representations. Since the discrete harmonic function approach is quite flexible it allows further applications, we mention some of them.

The drawback of the characterization of SLTRs is that we are not able to effectively check whether a given graph admits a flat angle assignment that fulfills the conditions. Hence it is still open to decide whether the recognition of graphs that admit straight line triangle representation is polynomially tractable.


Planar Graph Outer Face Outerplanar Graph Contact Family Convex Corner 
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  1. 1.
    Aerts, N., Felsner, S.: Henneberg steps for Triangle Representations,
  2. 2.
    Alam, M.J., Fowler, J., Kobourov, S.G.: Outerplanar graphs with proper touching triangle representations (unpublished manuscript)Google Scholar
  3. 3.
    de Fraysseix, H., de Mendez, P.O.: Stretching of jordan arc contact systems. In: Liotta, G. (ed.) GD 2003. LNCS, vol. 2912, pp. 71–85. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  4. 4.
    de Fraysseix, H., Ossona de Mendez, P.: Contact and intersection representations. In: Pach, J. (ed.) GD 2004. LNCS, vol. 3383, pp. 217–227. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  5. 5.
    de Fraysseix, H., de Mendez, P.O.: Barycentric systems and stretchability. Discrete Applied Mathematics 155, 1079–1095 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    de Verdière, Y.C.: Comment rendre géodésique une triangulation d’une surface? L’Enseignement Mathématique 37, 201–212 (1991)zbMATHGoogle Scholar
  7. 7.
    Gansner, E.R., Hu, Y., Kobourov, S.G.: On Touching Triangle Graphs. In: Brandes, U., Cornelsen, S. (eds.) GD 2010. LNCS, vol. 6502, pp. 250–261. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  8. 8.
    Haas, R., Orden, D., Rote, G., Santos, F., Servatius, B., Servatius, H., Souvaine, D.L., Streinu, I., Whiteley, W.: Planar minimally rigid graphs and pseudo-triangulations. Comput. Geom. 31, 31–61 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kenyon, R., Sheffield, S.: Dimers, tilings and trees. J. Comb. Theory, Ser. B 92, 295–317 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kobourov, S.G., Mondal, D., Nishat, R.I.: Touching triangle representations for 3-connected planar graphs. In: Didimo, W., Patrignani, M. (eds.) GD 2012. LNCS, vol. 7704, pp. 199–210. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  11. 11.
    Lovász, L.: Geometric representations of graphs, Draft version (December 11, 2009),
  12. 12.
    Tutte, W.T.: How to draw a graph. Proc. of the London Math. Society 13, 743–767 (1963)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  • Nieke Aerts
    • 1
  • Stefan Felsner
    • 1
  1. 1.Institut für MathematikTechnische Universität BerlinBerlinGermany

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