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Gromov Hyperbolicity in the Cartesian Sum of Graphs

  • W. Carballosa
  • A. de la Cruz
  • J. M. RodríguezEmail author
Original Paper

Abstract

In this paper, we characterize the hyperbolic product graphs for the Cartesian sum \(G_1\oplus G_2\): \(G_1\oplus G_2\) is always hyperbolic, unless either \(G_1\) or \(G_2\) is the trivial graph (the graph with a single vertex); if \(G_1\) or \(G_2\) is the trivial graph, then \(G_1\oplus G_2\) is hyperbolic if and only if \(G_2\) or \(G_1\) is hyperbolic, respectively. Besides, we characterize the Cartesian sums with hyperbolicity constant \(\delta (G_1\oplus G_2) = t\) for every value of t. Furthermore, we obtain the sharp inequalities \(1\le \delta (G_1\oplus G_2)\le 3/2\) for every non-trivial graphs \(G_1,G_2\). In addition, we obtain simple formulas for the hyperbolicity constant of the Cartesian sum of many graphs. Finally, we prove the inequalities \(3/2\le \delta (\overline{G_1\oplus G_2})\le 2\) for the complement graph of \(G_1\oplus G_2\) for every \(G_1,G_2\) with \(\min \{{{\mathrm{diam}}}V(G_1), {{\mathrm{diam}}}V(G_2)\}\ge 3\).

Keywords

Cartesian sum of graphs Geodesics Gromov hyperbolicity Complement of graphs 

Mathematics Subject Classification

Primary 05C75 Secondary 05C12 05A20 

Notes

Acknowledgements

We would like to thank the referee for his/her careful reading of this manuscript and some valuable comments which have improved the presentation of the paper. This work was supported in part by three grants from Ministerio de Economía y Competitividad (MTM2013-46374-P, MTM2016-78227-C2-1-P and MTM2015-69323-REDT), Spain, and a grant from CONACYT (FOMIX-CONACyT-UAGro 249818), México.

References

  1. 1.
    Abu-Ata, M., Dragan, F.F.: Metric tree-like structures in real-life networks: an empirical study. Networks 67(1), 49–68 (2016)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Adcock, A.B., Sullivan, B.D., Mahoney, M.W.: Tree-like structure in large social and information networks. In: 13th Int. Conference on Data Mining (ICDM), pp. 1–10. IEEE, Dallas, TX (2013)Google Scholar
  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: Group Theory from a Geometrical Viewpoint (Trieste, 1990), pp. 3–63, World Sci. Publ. River Edge, NJ (1991)Google Scholar
  4. 4.
    Bermudo, S., Carballosa, W., Rodríguez, J.M., Sigarreta, J.M.: On the hyperbolicity of edge-chordal and path-chordal graphs. Filomat 30(9), 2599–2607 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bermudo, S., Rodríguez, J.M., Sigarreta, J.M.: Computing the hyperbolicity constant. Comput. Math. Appl. 62(12), 4592–4595 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bermudo, S., Rodríguez, J.M., Sigarreta, J.M., Tourís, E.: Hyperbolicity and complement of graphs. Appl. Math. Lett. 24(11), 1882–1887 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bermudo, S., Rodríguez, J.M., Sigarreta, J.M., Vilaire, J.-M.: Gromov hyperbolic graphs. Discrete Math. 313(15), 1575–1585 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bowditch, B.H.: Notes on Gromov’s hyperbolicity criterion for path-metric spaces. Group Theory from a Geometrical Viewpoint (Trieste, 1990), pp. 64–67. World Science, River Edge (1991)Google Scholar
  9. 9.
    Brinkmann, G., Koolen, J., Moulton, V.: On the hyperbolicity of chordal graphs. Ann. Comb. 5(1), 61–69 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Calegari, D., Fujiwara, K.: Counting subgraphs in hyperbolic graphs with symmetry. J. Math. Soc. Jpn. 67(3), 1213–1226 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Carballosa, W., Casablanca, R.M., de la Cruz, A., Rodríguez, J.M.: Gromov hyperbolicity in strong product graphs. Electron. J. Combin. 20(3), 22 (2013)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Carballosa, W., de la Cruz, A., Rodríguez, J.M.: Gromov hyperbolicity in lexicographic product graphs. In: Proceedings Mathematical Sciences (2018)Google Scholar
  13. 13.
    Carballosa, W., Rodríguez, J.M., Sigarreta, J.M.: Hyperbolicity in the corona and join of graphs. Aequ. Math. 89(5), 1311–1327 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Carballosa, W., Rodríguez, J.M., Sigarreta, J.M., Villeta, M.: On the hyperbolicity constant of line graphs. Electron. J. Combin. 18(1), 18 (2011)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Charney, R.: Artin groups of finite type are biautomatic. Math. Ann. 292(4), 671–683 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Chen, B., Yau, S.-T., Yeh, Y.-N.: Graph homotopy and Graham homotopy. Discrete Math. 241(1–3), 153–170 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Chepoi, V., Dragan, F.F., Vaxès, Y.: Core congestion is inherent in hyperbolic networks. In: Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 2264–2279. SIAM, Philadelphia, PA (2017)Google Scholar
  18. 18.
    Čižek, N., Klavžar, S.: On the chromatic number of the lexicographic product and the Cartesian sum of graphs. Discrete Math. 134(1–3), 17–24 (1994)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Cohen, N., Coudert, D., Lancin, A.: Exact and approximate algorithms for computing the hyperbolicity of large-scale graphs. In: Rapport de Recherche 8074, INRIA, p. 28 (2012)Google Scholar
  20. 20.
    Coudert, D., Ducoffe, G.: Recognition of \(C_4\)-Free and \(1/2\)-Hyperbolic Graphs. SIAM J. Discrete Math. 28(1), 1601–1617 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Coudert, D., Ducoffe, G.: On the hyperbolicity of bipartite graphs and intersection graphs. Research Report, INRIA Sophia Antipolis-Méditerranée; I3S; Université Nice Sophia Antipolis; CNRS, p. 12 (2015)Google Scholar
  22. 22.
    Der-Fen Liu, D., Zhu, X.: Coloring the cartesian sum of graphs. Discrete Math. 308(24), 5928–5936 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Fournier, H., Ismail, A., Vigneron, A.: Computing the Gromov hyperbolicity of a discrete metric space. Inform. Process. Lett. 115(6–8), 576–579 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Ghys, E., de la Harpe, P.: Sur les Groupes Hyperboliques d’après Mikhael Gromov, Progress in Mathematics 83. Birkhäuser Boston Inc., Boston (1990)CrossRefzbMATHGoogle Scholar
  25. 25.
    Grippo, E., Jonckheere, E.A.: Effective resistance criterion for negative curvature: application to congestion control. In: Proceedings of 2016 IEEE Multi-Conference on Systems and Control, IEEE (2016)Google Scholar
  26. 26.
    Gromov, M.: Hyperbolic groups. Essays in Group Theory, Math. Sci. Res. Inst. Publ., vol. 8, pp. 75–263. Springer, New York (1987)CrossRefGoogle Scholar
  27. 27.
    Hammack, R., Imrich, W., Klavžar, S.: Handbook of Product Graphs, Discrete Mathematics and its Applications Series, 2nd edn. CRC, Boca Raton (2011)zbMATHGoogle Scholar
  28. 28.
    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(1), 45–60 (2002)Google Scholar
  29. 29.
    Jonckheere, E.A., Lohsoonthorn, P.: Geometry of network security. In: Proceedings of the 2004 American Control Conference, pp. 111–151. IEEE (2004)Google Scholar
  30. 30.
    Koolen, J.H., Moulton, V.: Hyperbolic bridged graphs. Eur. J. Combin. 23(6), 683–699 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Krioukov, D., Papadopoulos, F., Kitsak, M., Vahdat, A., Boguñá, M.: Hyperbolic geometry of complex networks. Phys. Rev. E 82(3), 18 (2010)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Kuziak, D.: Strong Resolvability in Product Graphs. PhD Thesis, Universitat Rovira I Virgili (2014)Google Scholar
  33. 33.
    Kuziak, D., Yero, I.G., Rodríguez-Velázquez, J.A.: On the strong metric dimension of the strong products of graphs. Open Math. 13(1), 64–74 (2015)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Li, S., Tucci, G.H.: Traffic congestion in expanders, \((p,\delta )\)-hyperbolic spaces and product of trees. Internet Math. 11(2), 134–142 (2015)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Martínez-Pérez, A.: Chordality properties and hyperbolicity on graphs. Electron. J. Combin. 23(3), 13 (2016)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Michel, J., Rodríguez, J.M., Sigarreta, J.M., Villeta, M.: Gromov hyperbolicity in Cartesian product graphs. Proc. Indian Acad. Sci. Math. Sci. 120(5), 1–17 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Michel, J., Rodríguez, J.M., Sigarreta, J.M., Villeta, M.: Hyperbolicity and parameters of graphs. Ars Combin. 100(1), 43–63 (2011)MathSciNetzbMATHGoogle Scholar
  38. 38.
    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. IEEE (2011)Google Scholar
  39. 39.
    Narayan, O., Saniee, I.: Large-scale curvature of networks. Phys. Rev. E 84(6), 066108 (2011)CrossRefGoogle Scholar
  40. 40.
    Nowakowski, R.J., Rall, D.F.: Associative graph products and their independence, domination and coloring numbers. Discuss. Math. Graph Theory 16(1), 53–79 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Ore, O.: Theory of Graphs. Amer. Math. Soc., Providence (1962)CrossRefzbMATHGoogle Scholar
  42. 42.
    Oshika, K.: Discrete Groups. Amer. Math. Soc., Providence (2002)Google Scholar
  43. 43.
    Papasoglu, P.: An algorithm detecting hyperbolicity, In: Geometric and Computational Perspectives on Infinite Groups, DIMACS Series in Discrete Mathematics and Theoretical Computer Science 25, pp. 193–200. Amer. Math. Soc. Providence, RI (1996)Google Scholar
  44. 44.
    Pestana, D., Rodríguez, J.M., Sigarreta, J.M., Villeta, M.: Gromov hyperbolic cubic graphs. Cent. Eur. J. Math. 10(3), 1141–1151 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Rodríguez, J.M., Sigarreta, J.M., Vilaire, J.-M., Villeta, M.: On the hyperbolicity constant in graphs. Discrete Math. 311(4), 211–219 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Scheinerman, E., Ullman, D.: Fractional Graph Theory, Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley, New York (1997)Google Scholar
  47. 47.
    Shang, Y.: Lack of Gromov-hyperbolicity in colored random networks. PanAm. Math. J. 21(1), 27–36 (2011)MathSciNetzbMATHGoogle Scholar
  48. 48.
    Shang, Y.: Lack of Gromov-hyperbolicity in small-world networks. Cent. Eur. J. Math. 10(3), 1152–1158 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Shang, Y.: Non-hyperbolicity of random graphs with given expected degrees. Stoch. Models 29(4), 451–462 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Shao, Z., Zhang, D.: The L(2,1)-labeling on Cartesian sum of graphs. Appl. Math. Lett. 21(8), 843–848 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Shavitt, Y., Tankel, T.: On internet embedding in hyperbolic spaces for overlay construction and distance estimation. INFOCOM, Cambridge (2004)Google Scholar
  52. 52.
    Tourís, E.: Graphs and Gromov hyperbolicity of non-constant negatively curved surfaces. J. Math. Anal. Appl. 380(2), 865–881 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    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)Google Scholar
  54. 54.
    Wu, Y., Zhang, C.: Chordality and hyperbolicity of a graph. Electron. J. Combin. 18(1), P43 (2011)zbMATHGoogle Scholar

Copyright information

© Iranian Mathematical Society 2018

Authors and Affiliations

  • W. Carballosa
    • 1
    • 2
  • A. de la Cruz
    • 3
  • J. M. Rodríguez
    • 3
    Email author
  1. 1.Department of Mathematics and StatisticsFlorida International UniversityMiamiUSA
  2. 2.Department of MathematicsMiami Dade CollegeMiamiUSA
  3. 3.Departamento de MatemáticasUniversidad Carlos III de MadridLeganésSpain

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