Acta Mathematica Sinica, English Series

, Volume 30, Issue 1, pp 79–90 | Cite as

Gromov hyperbolicity of periodic planar graphs

  • Alicia Cantón
  • Ana Granados
  • Domingo Pestana
  • José Manuel Rodríguez


The study of hyperbolic graphs is an interesting topic since the hyperbolicity of a geodesic metric space is equivalent to the hyperbolicity of a graph related to it. The main result in this paper is a very simple characterization of the hyperbolicity of a large class of periodic planar graphs.


Planar graphs periodic graphs Gromov hyperbolicity infinite graphs geodesics 

MR(2010) Subject Classification

05C10 05C63 05C75 05A20 


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  1. [1]
    Alonso, J., Brady, T., Cooper, D., et al.: Notes on word hyperbolic groups. In: Group Theory from a Geometrical Viewpoint, E. Ghys, A. Haefliger, A. Verjovsky Eds., World Scientific, Singapore, 1992Google Scholar
  2. [2]
    Bermudo, S., Rodríguez, J. M., Sigarreta, J. M.: Computing the hyperbolicity constant. Comput. Math. Appl., 62, 4592–4595 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Bermudo, S., Rodríguez, J. M., Sigarreta, J. M., et al.: Hyperbolicity and complement of graphs. Appl. Math. Lett., 24, 1882–1887 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Bowditch, B. H.: Notes on Gromov’s hyperbolicity criterion for path-metric spaces. Group theory from a geometrical viewpoint, Trieste, 1990 (ed. E. Ghys, A. Haefliger and A. Verjovsky; World Scientific, River Edge NJ, 1991, 64–167)Google Scholar
  5. [5]
    Brinkmann, G., Koolen J., Moulton, V.: On the hyperbolicity of chordal graphs. Ann. Comb., 5, 61–69 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Carballosa, W., Pestana, D., Rodríguez, J. M., et al.: Distortion of the hyperbolicity constant of a graph. Electr. J. Comb., 19, p67 (2012)CrossRefGoogle Scholar
  7. [7]
    Carballosa, W., Rodríguez, J. M., Sigarreta, J. M., et al.: Gromov hyperbolicity of line graphs. Electr. J. Comb., 18, p210 (2011)Google Scholar
  8. [8]
    Chen, B., Yau, S.-T., Yeh, Y.-N.: Graph homotopy and Graham homotopy. Discret. Math., 241, 153–170 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Chepoi, V., Estellon, B.: Packing and covering δ-hyperbolic spaces by balls. APPROX-RANDOM 2007, 59–73Google Scholar
  10. [10]
    Eppstein, D.: Squarepants in a tree: sum of subtree clustering and hyperbolic pants decomposition. SODA 2007, 29–38Google Scholar
  11. [11]
    Gavoille, C., Ly, O.: Distance labeling in hyperbolic graphs. ISAAC 2005, Volume LNCS 3827, 1071–1079Google Scholar
  12. [12]
    Ghys, E., de la Harpe, P.: Sur les Groupes Hyperboliques d’après Mikhael Gromov, Progress in Mathematics 83, Birkhäuser Boston Inc., Boston, MA, 1990Google Scholar
  13. [13]
    Gromov, M.: Hyperbolic groups. In: Essays in Group Theory, Edited by S. M. Gersten, M. S. R. I. Publ., 8, Springer, 1987, 75–263CrossRefGoogle Scholar
  14. [14]
    Jonckheere, E. A.: Contrôle du traffic sur les réseaux à géométrie hyperbolique-Vers une théorie géométrique de la sécurité l’acheminement de l’information. J. Europ. Syst. Autom., 8, 45–60 (2002)Google Scholar
  15. [15]
    Jonckheere, E., Lohsoonthorn, P.: A hyperbolic geometry approach to multipath routing. In: Proceedings of the 10th Mediterranean Conference on Control and Automation (MED 2002), Lisbon, Portugal, July 2002, FA5-1Google Scholar
  16. [16]
    Jonckheere, E. A., Lohsoonthorn, P.: Geometry of network security. Amer. Control Conf., 6, 976–981 (2004)Google Scholar
  17. [17]
    Krauthgamer, R., Lee, J. R.: Algorithms on negatively curved spaces. FOCS 2006Google Scholar
  18. [18]
    Koolen, J. H., Moulton, V.: Hyperbolic Bridged Graphs. Europ. J. Combin., 23, 683–699 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    Michel, J., Rodríguez, J. M., Sigarreta, J. M., et al.: Hyperbolicity and parameters of graphs. Ars Comb., Volume C, 43–63 (2011)Google Scholar
  20. [20]
    Oshika, K.: Discrete Groups, AMS Bookstore, Providence, RI, 2002Google Scholar
  21. [21]
    Portilla, A., Rodríguez, J. M., Tourís, E.: Gromov hyperbolicity through decomposition of metric spaces II. J. Geom. Anal., 14, 123–149 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    Rodríguez, J. M., Sigarreta, J. M., et al., Villeta, M.: On the hyperbolicity constant in graphs. Discret. Math., 311, 211–219 (2011)CrossRefzbMATHGoogle Scholar
  23. [23]
    Rodríguez, J. M., Tourís, E.: Gromov hyperbolicity of Riemann surfaces. Acta Math. Sin., Engl. Series, 23(2), 209–228 (2007)CrossRefzbMATHGoogle Scholar
  24. [24]
    Shavitt, Y., Tankel, T.: On internet embedding in hyperbolic spaces for overlay construction and distance estimation. INFOCOM 2004Google Scholar
  25. [25]
    Tourís, E.: Graphs and Gromov hyperbolicity of non-constant negatively curved surfaces. J. Math. Anal. Appl., 380, 865–881 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    Wu, Y., Zhang, C.: Chordality and hyperbolicity of a graph. Electr. J. Combin., 18, p43 (2011)Google Scholar

Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Alicia Cantón
    • 1
  • Ana Granados
    • 2
  • Domingo Pestana
    • 3
  • José Manuel Rodríguez
    • 3
  1. 1.Departamento de Ciencias Aplicadas a la Ingeniería Naval, ETSINUniversidad Politécnica de MadridMadridSpain
  2. 2.Mathematics DivisionSt. Louis University (Madrid Campus)MadridSpain
  3. 3.Departamento de MatemáticasUniversidad Carlos III de MadridLeganés, MadridSpain

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