Advertisement

Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Planarity and Hyperbolicity in Graphs

Abstract

If X is a geodesic metric space and \(x_1,x_2,x_3\) are three points in \(X\), a geodesic triangle \(T=\{x_1,x_2,x_3\}\) is the union of three geodesics \([x_1x_2]\), \([x_2x_3]\) and \([x_3x_1]\) in \(X\). The space \(X\) is \(\delta \)-hyperbolic \((\)in the Gromov sense\()\) if any side of \(T\) is contained in a \(\delta \)-neighborhood of the union of the two other sides, for every geodesic triangle \(T\) in \(X\). 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. In this paper we obtain criteria which allow us to decide, for a large class of graphs, whether they are hyperbolic or not. We are especially interested in the planar graphs which are the “boundary” (the \(1\)-skeleton) of a tessellation of the Euclidean plane. Furthermore, we prove that a graph obtained as the \(1\)-skeleton of a general CW \(2\)-complex is hyperbolic if and only if its dual graph is hyperbolic.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    Balogh, Z.M., Buckley, S.M.: Geometric characterizations of Gromov hyperbolicity. Invent. Math. 153, 261–301 (2003)

  2. 2.

    Boguna, M., Papadopoulos, F., Krioukov, D.: Sustaining the Internet with hyperbolic mapping. Nat. Commun. 1(62), 18 (2010)

  3. 3.

    Bermudo, S., Rodríguez, J.M., Sigarreta, J.M.: Computing the hyperbolicity constant. Comput. Math. Appl. 62(12), 4592–4595 (2011)

  4. 4.

    Bermudo, S., Rodríguez, J.M., Sigarreta, J.M., Tourís, E.: Hyperbolicity and complement of graphs. Appl. Math. Lett. 24, 1882–1887 (2011)

  5. 5.

    Bermudo, S., Rodríguez, J.M., Sigarreta, J.M., Vilaire, J.-M.: Gromov hyperbolic graphs. Discr. Math. 313(15), 1575–1585 (2013)

  6. 6.

    Bonk, M., Heinonen, J., Koskela, P., Uniformizing Gromov hyperbolic spaces. Astérisque 270 (2001)

  7. 7.

    Carballosa, W., Rodríguez, J.M., Sigarreta, J.M.: Distortion of the hyperbolicity constant of a graph. Electr. J. Comb. 19, P67 (2012)

  8. 8.

    Carballosa, W., Rodríguez, J.M., Sigarreta, J.M., Villeta, M.: On the hyperbolicity constant of line graphs. Electr. J. Comb. 18, P210 (2011)

  9. 9.

    Frigerio, R., Sisto, A.: Characterizing hyperbolic spaces and real trees. Geom. Dedicata 142, 139–149 (2009)

  10. 10.

    Ghys, E., de la Harpe, P.: Sur les Groupes Hyperboliques d’après Mikhael Gromov. Progress in Mathematics 83, Birkhäuser Boston Inc., Boston, MA (1990)

  11. 11.

    Gromov, M.: Hyperbolic groups. In: Gersten, S.M. (ed.) Essays in group theory. MSRI Publ. 8. Springer, pp. 75–263 (1987)

  12. 12.

    Hästö, P.A.: Gromov hyperbolicity of the \(j_G\) and \(\tilde{j}_G\) metrics. Proc. Am. Math. Soc. 134, 1137–1142 (2006)

  13. 13.

    Hästö, P.A., Portilla, A., Rodríguez, J.M., Tourís, E.: Gromov hyperbolic equivalence of the hyperbolic and quasihyperbolic metrics in Denjoy domains. Bull. Lond. Math. Soc. 42, 282–294 (2010)

  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)

  15. 15.

    Jonckheere, E.A., Lohsoonthorn, P.: Geometry of network security. Am. Control Conf. ACC, 111–151 (2004)

  16. 16.

    Michel, J., Rodríguez, J.M., Sigarreta, J.M. Villeta, M.: Hyperbolicity and parameters of graphs. Ars Comb. Volume C, 43–63 (2011)

  17. 17.

    Oshika, K.: Discrete groups, AMS Bookstore (2002)

  18. 18.

    Pestana, D., Rodríguez, J.M., Sigarreta, J.M., Villeta, M.: Gromov hyperbolic cubic graphs. Central Euro. J. Math. 10(3), 1141–1151 (2012)

  19. 19.

    Portilla, A., Rodríguez, J. M., Sigarreta, J. M. and Vilaire, J.-M.: Gromov hyperbolic tessellation graphs, to appear in Utilitas Math. Preprint in http://gama.uc3m.es/index.php/jomaro.html

  20. 20.

    Portilla, A., Rodríguez, J.M., Tourís, E.: Gromov hyperbolicity through decomposition of metric spaces II. J. Geom. Anal. 14, 123–149 (2004)

  21. 21.

    Portilla, A., Tourís, E.: A characterization of Gromov hyperbolicity of surfaces with variable negative curvature. Publ. Mat. 53, 83–110 (2009)

  22. 22.

    Rodríguez, J.M., Sigarreta, J.M.: Bounds on Gromov hyperbolicity constant in graphs. Proc. Indian Acad. Sci. Math. Sci. 122, 53–65 (2012)

  23. 23.

    Rodríguez, J.M., Sigarreta, J.M., Vilaire, J.-M., Villeta, M.: On the hyperbolicity constant in graphs. Discr. Math. 311, 211–219 (2011)

  24. 24.

    Tourís, E.: Graphs and Gromov hyperbolicity of non-constant negatively curved surfaces. J. Math. Anal. Appl. 380, 865–881 (2011)

  25. 25.

    Wu, Y., Zhang, C.: Chordality and hyperbolicity of a graph. Electr. J. Comb. 18, P43 (2011)

Download references

Acknowledgments

This work was partly supported by the Spanish Ministry of Science and Innovation through projects MTM 2009-07800, MTM 2009-12740-C03-01 and MTM 2008-02829-E.

Author information

Correspondence to Walter Carballosa.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Carballosa, W., Portilla, A., Rodríguez, J.M. et al. Planarity and Hyperbolicity in Graphs. Graphs and Combinatorics 31, 1311–1324 (2015). https://doi.org/10.1007/s00373-014-1459-4

Download citation

Keywords

  • Tessellation
  • Planar graph
  • Gromov hyperbolicity
  • CW complex
  • Dual graph

Mathematics Subject Classification

  • 05C69
  • 05A20
  • 05C50