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Gromov Hyperbolicity in the Cartesian Sum of Graphs

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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 \(\min \{{{\mathrm{diam}}}V(G_1), {{\mathrm{diam}}}V(G_2)\}\ge 3\).

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References

  1. Abu-Ata, M., Dragan, F.F.: Metric tree-like structures in real-life networks: an empirical study. Networks 67(1), 49–68 (2016)

    Article  MathSciNet  Google Scholar 

  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)

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Bermudo, S., Rodríguez, J.M., Sigarreta, J.M., Tourís, E.: Hyperbolicity and complement of graphs. Appl. Math. Lett. 24(11), 1882–1887 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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. Brinkmann, G., Koolen, J., Moulton, V.: On the hyperbolicity of chordal graphs. Ann. Comb. 5(1), 61–69 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  10. Calegari, D., Fujiwara, K.: Counting subgraphs in hyperbolic graphs with symmetry. J. Math. Soc. Jpn. 67(3), 1213–1226 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  12. Carballosa, W., de la Cruz, A., Rodríguez, J.M.: Gromov hyperbolicity in lexicographic product graphs. In: Proceedings Mathematical Sciences (2018)

  13. Carballosa, W., Rodríguez, J.M., Sigarreta, J.M.: Hyperbolicity in the corona and join of graphs. Aequ. Math. 89(5), 1311–1327 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  15. Charney, R.: Artin groups of finite type are biautomatic. Math. Ann. 292(4), 671–683 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  16. Chen, B., Yau, S.-T., Yeh, Y.-N.: Graph homotopy and Graham homotopy. Discrete Math. 241(1–3), 153–170 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

    MathSciNet  MATH  Google Scholar 

  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)

  20. Coudert, D., Ducoffe, G.: Recognition of \(C_4\)-Free and \(1/2\)-Hyperbolic Graphs. SIAM J. Discrete Math. 28(1), 1601–1617 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  22. Der-Fen Liu, D., Zhu, X.: Coloring the cartesian sum of graphs. Discrete Math. 308(24), 5928–5936 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  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)

  26. Gromov, M.: Hyperbolic groups. Essays in Group Theory, Math. Sci. Res. Inst. Publ., vol. 8, pp. 75–263. Springer, New York (1987)

    Chapter  Google Scholar 

  27. Hammack, R., Imrich, W., Klavžar, S.: Handbook of Product Graphs, Discrete Mathematics and its Applications Series, 2nd edn. CRC, Boca Raton (2011)

    MATH  Google Scholar 

  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. Jonckheere, E.A., Lohsoonthorn, P.: Geometry of network security. In: Proceedings of the 2004 American Control Conference, pp. 111–151. IEEE (2004)

  30. Koolen, J.H., Moulton, V.: Hyperbolic bridged graphs. Eur. J. Combin. 23(6), 683–699 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  31. Krioukov, D., Papadopoulos, F., Kitsak, M., Vahdat, A., Boguñá, M.: Hyperbolic geometry of complex networks. Phys. Rev. E 82(3), 18 (2010)

    Article  MathSciNet  Google Scholar 

  32. Kuziak, D.: Strong Resolvability in Product Graphs. PhD Thesis, Universitat Rovira I Virgili (2014)

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  35. Martínez-Pérez, A.: Chordality properties and hyperbolicity on graphs. Electron. J. Combin. 23(3), 13 (2016)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  37. Michel, J., Rodríguez, J.M., Sigarreta, J.M., Villeta, M.: Hyperbolicity and parameters of graphs. Ars Combin. 100(1), 43–63 (2011)

    MathSciNet  MATH  Google Scholar 

  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)

  39. Narayan, O., Saniee, I.: Large-scale curvature of networks. Phys. Rev. E 84(6), 066108 (2011)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  41. Ore, O.: Theory of Graphs. Amer. Math. Soc., Providence (1962)

    Book  MATH  Google Scholar 

  42. Oshika, K.: Discrete Groups. Amer. Math. Soc., Providence (2002)

    Google Scholar 

  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)

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

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  46. Scheinerman, E., Ullman, D.: Fractional Graph Theory, Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley, New York (1997)

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  50. Shao, Z., Zhang, D.: The L(2,1)-labeling on Cartesian sum of graphs. Appl. Math. Lett. 21(8), 843–848 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  51. Shavitt, Y., Tankel, T.: On internet embedding in hyperbolic spaces for overlay construction and distance estimation. INFOCOM, Cambridge (2004)

    Google Scholar 

  52. Tourís, E.: Graphs and Gromov hyperbolicity of non-constant negatively curved surfaces. J. Math. Anal. Appl. 380(2), 865–881 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  54. Wu, Y., Zhang, C.: Chordality and hyperbolicity of a graph. Electron. J. Combin. 18(1), P43 (2011)

    MATH  Google Scholar 

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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.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Florida International University, 11200 SW 8th Street, Miami, FL, 33199, USA

    W. Carballosa

  2. Department of Mathematics, Miami Dade College, 300 NE Second Ave., Miami, FL, 33132, USA

    W. Carballosa

  3. Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911, Leganés, Madrid, Spain

    A. de la Cruz & J. M. Rodríguez

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  1. W. Carballosa
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  2. A. de la Cruz
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  3. J. M. Rodríguez
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Correspondence to J. M. Rodríguez.

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Communicated by Hossein Hajiabolhassan.

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Carballosa, W., de la Cruz, A. & Rodríguez, J.M. Gromov Hyperbolicity in the Cartesian Sum of Graphs. Bull. Iran. Math. Soc. 44, 837–856 (2018). https://doi.org/10.1007/s41980-018-0055-4

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