Touching Triangle Representations for 3-Connected Planar Graphs

  • Stephen G. Kobourov
  • Debajyoti Mondal
  • Rahnuma Islam Nishat
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7704)


A touching triangle graph (TTG) representation of a planar graph is a planar drawing Γ of the graph, where each vertex is represented as a triangle and each edge e is represented as a side contact of the triangles that correspond to the end vertices of e. We call Γ a proper TTG representation if Γ determines a tiling of a triangle, where each tile corresponds to a distinct vertex of the input graph. In this paper we prove that every 3-connected cubic planar graph admits a proper TTG representation. We also construct proper TTG representations for parabolic grid graphs and the graphs determined by rectangular grid drawings (e.g., square grid graphs). Finally, we describe a fixed-parameter tractable decision algorithm for testing whether a 3-connected planar graph admits a proper TTG representation.


Planar Graph Input Graph Grid Graph Path Covering Planar Embedding 
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.
    Batagelj, V.: Inductive classes of cubic graphs. In: Proceedings of the 6th Hungarian Colloquium on Combinatorics, Eger, Hungary. Finite and infinite sets, vol. 37, pp. 89–101 (1981)Google Scholar
  2. 2.
    Buchsbaum, A., Gansner, E., Procopiuc, C., Venkatasubramanian, S.: Rectangular layouts and contact graphs. ACM Transactions on Algorithms 4(1) (2008)Google Scholar
  3. 3.
    de Fraysseix, H., de Mendez, P.O.: Barycentric systems and stretchability. Discrete Applied Mathematics 155(9), 1079–1095 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    de Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10, 41–51 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    de Fraysseix, H., de Mendez, P.O., Rosenstiehl, P.: On triangle contact graphs. Combinatorics, Probability & Computing 3, 233–246 (1994)zbMATHCrossRefGoogle Scholar
  6. 6.
    Duncan, C., Gansner, E.R., Hu, Y., Kaufmann, M., Kobourov, S.G.: Optimal polygonal representation of planar graphs. Algorithmica 63(3), 672–691 (2012)MathSciNetzbMATHCrossRefGoogle 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.
    He, X.: On floor-plan of plane graphs. SIAM Journal on Computing 28(6), 2150–2167 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Koebe, P.: Kontaktprobleme der konformen Abbildung. Berichte über die Verhandlungen der Sächsischen Akademie der Wissenschaften zu Leipzig. Math.-Phys. Klasse 88, 141–164 (1936)Google Scholar
  10. 10.
    Liao, C.C., Lu, H.I., Yen, H.C.: Compact floor-planning via orderly spanning trees. Journal of Algorithms 48, 441–451 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Phillips, R.: The Order-5 triangle partitions, (accessed June 7, 2012)
  12. 12.
    Rahman, M., Nishizeki, T., Ghosh, S.: Rectangular drawings of planar graphs. Journal of Algorithms 50(1), 62–78 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Thomassen, C.: Interval representations of planar graphs. Journal of Combinatorial Theory (B) 40(1), 9–20 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Woeginger, G.J.: Exact Algorithms for NP-Hard Problems: A Survey. In: Jünger, M., Reinelt, G., Rinaldi, G. (eds.) Combinatorial Optimization - Eureka, You Shrink! LNCS, vol. 2570, pp. 185–207. Springer, Heidelberg (2003)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Stephen G. Kobourov
    • 1
  • Debajyoti Mondal
    • 2
  • Rahnuma Islam Nishat
    • 3
  1. 1.Department of Computer ScienceUniversity of ArizonaUSA
  2. 2.Department of Computer ScienceUniversity of ManitobaCanada
  3. 3.Department of Computer ScienceUniversity of VictoriaCanada

Personalised recommendations