On the Hyperbolicity Constant in Graph Minors

Abstract

A graph H is a minor of a graph G if a graph isomorphic to H can be obtained from G by contracting some edges, deleting some edges, and deleting some isolated vertices. 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. One of the main aims in this work is to obtain quantitative information about the distortion of the hyperbolicity constant of the graph \(\,G/e\) obtained from the (simple or non-simple) graph G by contracting an arbitrary edge e from it. We prove that H is hyperbolic if and only if G is hyperbolic, for many minors H of G, even if H is obtained from G by contracting and/or deleting infinitely many edges.

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

References

  1. 1.

    Abu-Ata, M., Dragan, F.F.: Metric tree-like structures in real-life networks: an empirical study. Networks 67, 49–68 (2016)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Adcock, A.B., Sullivan, B.D., Mahoney, M.W.: Tree-like structure in large social and information networks, 13th Int Conference Data Mining (ICDM), pp. 1–10. Dallas, Texas, USA, IEEE (2013)

  3. 3.

    Alonso, J., Brady, T., Cooper, D., Delzant, T., Ferlini, V., Lustig, M., Mihalik, M., Shapiro, M., Short, H.: Notes on word hyperbolic groups. In: Ghys, E., Haefliger, A., Verjovsky, A. (eds.) Group Theory from a Geometrical Viewpoint. World Scientific, Singapore (1992)

    Google Scholar 

  4. 4.

    Bandelt, H.-J., Chepoi, V.: \(1\)-Hyperbolic graphs. SIAM J. Discr. Math. 16, 323–334 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

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

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

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

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Birmelé, E., Bondy, J.A., Reed, B.A.: Tree-width of graphs without a \(3 \times 3\) grid minor. Discr. Appl. Math. 157, 2577–2596 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Bowditch, B.H.: Notes on Gromov’s hyperbolicity criterion for path-metric spaces in Group theory from a geometrical viewpoint, Trieste, 1990 (ed. E. Ghys, A. Haefliger and A. Verjovsky; World Scientific, River Edge, NJ, 1991), pp. 64–167 (1990)

  9. 9.

    Brinkmann, G., Koolen, J., Moulton, V.: On the hyperbolicity of chordal graphs. Ann. Comb. 5, 61–69 (2001)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Calegari, D., Fujiwara, K.: Counting subgraphs in hyperbolic graphs with symmetry. J. Math. Soc. Jpn. 67, 1213–1226 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Carballosa, W., Pestana, D., Rodríguez, J.M., Sigarreta, J.M.: Distortion of the hyperbolicity constant of a graph. Electr. J. Comb. 19, 67 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Carballosa, W., Rodríguez, J.M., Sigarreta, J.M.: Hyperbolicity in the corona and join graphs. Aequat. Math. 89, 1311–1327 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Charney, R.: Artin groups of finite type are biautomatic. Math. Ann. 292, 671–683 (1992)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Chen, W., Fang, W., Hu, G., Mahoney, M.W.: On the hyperbolicity of small-world and treelike random graphs. Int. Math. 9(4), 434–491 (2013)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Chen, B., Yau, S.-T., Yeh, Y.-N.: Graph homotopy and Graham homotopy. Discr. Math. 241, 153–170 (2001)

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Diestel, R., Müller, M.: Connected tree-width. Combinatorica, First Online: 13 February (2017). https://doi.org/10.1007/s00493-016-3516-5

  17. 17.

    Eppstein, D.: Squarepants in a tree: sum of subtree clustering and hyperbolic pants decomposition, SODA’2007. In Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 29–38 (2007)

  18. 18.

    Fournier, H., Ismail, A., Vigneron, A.: Computing the Gromov hyperbolicity of a discrete metric space. J. Inf. Proc. Lett. 115, 576–579 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Gavoille, C., Ly, O.: Distance labeling in hyperbolic graphs. In ISAAC, pp. 171–179 (2005)

  20. 20.

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

  21. 21.

    Grigoriev, A., Marchal, B., Usotskaya, N.: On planar graphs with large tree-width and small grid minors. Electr. Notes Discr. Math. 32, 35–42 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    Gromov, M.: Hyperbolic groups. In: “Essays in group theory”. Edited by S. M. Gersten, M. S. R. I. Publ. 8. Springer, 75–263 (1987)

  23. 23.

    Hamann, M.: On the tree-likeness of hyperbolic spaces (2011). arXiv:1105.3925

  24. 24.

    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. FA5-1 (2002)

  25. 25.

    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. Eur. Syst. Autom. 8, 45–60 (2002)

    Google Scholar 

  26. 26.

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

  27. 27.

    Koolen, J.H., Moulton, V.: Hyperbolic bridged graphs. Eur. J. Comb. 23, 683–699 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  28. 28.

    Krauthgamer, R., Lee. J.R.: Algorithms on negatively curved spaces. FOCS (2006)

  29. 29.

    Krioukov, D., Papadopoulos, F., Kitsak, M., Vahdat, A., Boguñá, M.: Hyperbolic geometry of complex networks. Phys. Rev. E 82, 036106 (2010)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Kuratowski, K.: Sur le problème des courbes gauches en topologie (in French). Fund. Math. 15, 271–283 (1930)

    Article  MATH  Google Scholar 

  31. 31.

    Li, S., Tucci, G.H.: Traffic congestion in expanders, \((p, \delta )\)-hyperbolic spaces and product of trees. Int. Math. 11(2), 134–142 (2015)

    Google Scholar 

  32. 32.

    Martínez-Pérez, A.: Chordality properties and hyperbolicity on graphs. Electr. J. Comb. 23:3, P3.51 (2016)

    MathSciNet  MATH  Google Scholar 

  33. 33.

    Martínez-Pérez, A.: Generalized chordality, vertex separators and hyperbolicity on graphs. Symmetry 10, 199 (2017). https://doi.org/10.3390/sym9100199.

  34. 34.

    Michel, J., Rodríguez, J.M., Sigarreta, J.M., Villeta, M.: Hyperbolicity and parameters of graphs. ARS Comb. 100, 43–63 (2011)

    MathSciNet  MATH  Google Scholar 

  35. 35.

    Montgolfier, F., Soto, M., Viennot, L.: Treewidth and hyperbolicity of the internet. In: 10th IEEE International Symposium on Network Computing and Applications (NCA), pp. 25–32 (2011)

  36. 36.

    Narayan, O., Saniee, I.: Large-scale curvature of networks. Phys. Rev. E 84, 066108 (2011)

    Article  Google Scholar 

  37. 37.

    Oshika, K.: Discrete Groups. AMS Bookstore, New York (2002)

    Google Scholar 

  38. 38.

    Papasoglu, P.: An algorithm detecting hyperbolicity, in Geometric and computational perspectives on infinite groups. DIMACS - Series in Discrete Mathematics and Theoretical Computer Science Volume 25, AMS, pp. 193–200 (1996)

  39. 39.

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

    MathSciNet  Article  MATH  Google Scholar 

  40. 40.

    Portilla, A., Rodríguez, J.M., Sigarreta, J.M., Vilaire, J.-M.: Gromov hyperbolic tessellation graphs. Utilitas Math. 97, 193–212 (2015)

    MathSciNet  MATH  Google Scholar 

  41. 41.

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

    MathSciNet  Article  MATH  Google Scholar 

  42. 42.

    Robertson, N., Seymour, P.D.: Graph minors. III. Planar tree-width. J. Combin. Theory Ser. B 36, 49–64 (1984)

    MathSciNet  Article  MATH  Google Scholar 

  43. 43.

    Robertson, N., Seymour, P.D.: Graph minors. X. Obstructions to tree-decomposition. J. Combin. Theory Ser. B 52, 153–190 (1991)

    MathSciNet  Article  MATH  Google Scholar 

  44. 44.

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

    MathSciNet  Article  MATH  Google Scholar 

  45. 45.

    Shang, Y.: Lack of Gromov-hyperbolicity in colored random networks. Pan-Am. Math. J. 21(1), 27–36 (2011)

    MathSciNet  MATH  Google Scholar 

  46. 46.

    Shang, Y.: Lack of Gromov-hyperbolicity in small-world networks. Cent. Eur. J. Math. 10(3), 1152–1158 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  47. 47.

    Shang, Y.: Non-hyperbolicity of random graphs with given expected degrees. Stoch. Models 29(4), 451–462 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  48. 48.

    Shavitt, Y., Tankel, T.: On internet embedding in hyperbolic spaces for overlay construction and distance estimation. IEEE/ACM Trans. Netw. 16(1), 25–36 (2008)

    Article  Google Scholar 

  49. 49.

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

    MathSciNet  Article  MATH  Google Scholar 

  50. 50.

    Verbeek, K., Suri, S.: Metric embeddings, hyperbolic space and social networks. In: Proceedings of the 30th Annual Symposium on Computational Geometry, pp. 501–510 (2014)

  51. 51.

    Wagner, K.: Über eine Eigenschaft der ebenen Komplexe. Math. Ann. 114, 570–590 (1937)

    MathSciNet  Article  MATH  Google Scholar 

  52. 52.

    Wang, C., Jonckheere, E., Brun, T.: Ollivier-Ricci curvature and fast approximation to tree-width in embeddability of QUBO problems. In: Proceedings of 6th International Symposium on Communications, Control, and Signal Processing (ISCCSP) (2014)

  53. 53.

    Wu, Y., Zhang, C.: Chordality and hyperbolicity of a graph. Electr. J. Comb. 18, 43 (2011)

    MATH  Google Scholar 

Download references

Acknowledgements

This work was supported in part by two grants from Ministerio de Economía y Competititvidad (MTM2013-46374-P and MTM2015-69323-REDT), Spain, and a Grant from CONACYT (FOMIX-CONACyT-UAGro 249818), México. Additionally, we would like to thank the referee for him/her careful reading of the manuscript and several useful comments which have helped us to improve the presentation of this paper.

Author information

Affiliations

Authors

Corresponding author

Correspondence to José M. Rodríguez.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Carballosa, W., Rodríguez, J.M., Rosario, O. et al. On the Hyperbolicity Constant in Graph Minors. Bull. Iran. Math. Soc. 44, 481–503 (2018). https://doi.org/10.1007/s41980-018-0032-y

Download citation

Keywords

  • Graph minor
  • Edge contraction
  • Deleted edge
  • Hyperbolic graph
  • Geodesics

Mathematics Subject Classification

  • Primary 05C75
  • Secondary 05C63
  • 05A20