# Gromov hyperbolicity in lexicographic product graphs

Article

## Abstract

If X is a geodesic metric space and $$x_1,x_2,x_3\in X$$, a geodesic triangle $$T=\{x_1,x_2,x_3\}$$ is the union of the 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. If X is hyperbolic, we denote by $$\delta (X)$$ the sharp hyperbolicity constant of X, i.e. $$\delta (X)=\inf \{\delta \ge 0: \, X \, \text { is }\delta \text {-hyperbolic}\}$$. In this paper, we characterize the lexicographic product of two graphs $$G_1\circ G_2$$ which are hyperbolic, in terms of $$G_1$$ and $$G_2$$: the lexicographic product graph $$G_1\circ G_2$$ is hyperbolic if and only if $$G_1$$ is hyperbolic, unless if $$G_1$$ is a trivial graph (the graph with a single vertex); if $$G_1$$ is trivial, then $$G_1\circ G_2$$ is hyperbolic if and only if $$G_2$$ is hyperbolic. In particular, we obtain the sharp inequalities $$\delta (G_1)\le \delta (G_1\circ G_2) \le \delta (G_1) + 3/2$$ if $$G_1$$ is not a trivial graph, and we characterize the graphs for which the second inequality is attained.

## Keywords

Lexicographic product graphs geodesics Gromov hyperbolicity infinite graphs

## Mathematics Subject Classification

Primary: 05C76 05C10 Secondary: 05C35 05C63 05C12

## References

1. 1.
Alonso J, Brady T, Cooper D, Delzant T, Ferlini V, Lustig M, Mihalik M, Shapiro M and Short H, Notes on word hyperbolic groups, in: Group Theory from a Geometrical Viewpoint edited by E Ghys, A Haefliger and A Verjovsky (1992) (Singapore: World Scientific)Google Scholar
2. 2.
Alvarez V, Portilla A, Rodríguez J M and Tourís E, Gromov hyperbolicity of Denjoy domains, Geom. Dedicata 121 (2006) 221–245
3. 3.
Balogh Z M and Bonk M, Gromov hyperbolicity and the Kobayashi metric on strictly pseudoconvex domains, Comm. Math. Helv. 75 (2000) 504–533
4. 4.
Balogh Z M and Buckley S M, Geometric characterizations of Gromov hyperbolicity, Invent. Math. 153 (2003) 261–301
5. 5.
Benoist Y, Convexes hyperboliques et fonctions quasisymétriques, Publ. Math. Inst. Hautes Études Sci. 97 (2003) 181–237
6. 6.
Bermudo S, Rodríguez J M, Rosario O and Sigarreta J M, Small values of the hyperbolicity constant in graphs, Discrete Math. 339 (2016) 3073–3084
7. 7.
Bermudo S, Rodríguez J M and Sigarreta J M, Computing the hyperbolicity constant, Comput. Math. Appl. 62 (2011) 4592–4595
8. 8.
Bermudo S, Rodríguez J M, Sigarreta J M and Vilaire J-M, Gromov hyperbolic graphs, Discrete Math. 313 (2013) 1575–1585
9. 9.
Bermudo S, Rodríguez J M, Sigarreta J M and Tourís E, Hyperbolicity and complement of graphs, Appl. Math. Lett. 24 (2011) 1882–1887
10. 10.
Bonk M, Heinonen J and Koskela P, Uniformizing Gromov hyperbolic spaces, Astérisque (2001) vol. 270Google Scholar
11. 11.
Brinkmann G, Koolen J and Moulton V, On the hyperbolicity of chordal graphs, Ann. Comb. 5 (2001) 61–69
12. 12.
Carballosa W, Casablanca R M, de la Cruz A and Rodríguez J M, Gromov hyperbolicity in strong product graphs, Electr. J. Comb. 20(3) (2013) P2
13. 13.
Carballosa W, Pestana D, Rodríguez J M and Sigarreta J M, Distortion of the hyperbolicity constant of a graph, Electr. J. Comb. 19 (2012) P67
14. 14.
Carballosa W, Rodríguez J M and Sigarreta J M, New inequalities on the hyperbolicity constant of line graphs, Ars Combin. 129 (2016) 367–386
15. 15.
Carballosa W, Rodríguez J M and Sigarreta J M, Hyperbolicity in the corona and join of graphs, Aequat. Math. 89 (2015) 1311–1327
16. 16.
Carballosa W, Rodríguez J M, Sigarreta J M and Villeta M, Gromov hyperbolicity of line graphs, Electr. J. Comb. 18 (2011) P210
17. 17.
Charney R, Artin groups of finite type are biautomatic, Math. Ann. 292 (1992) 671–683
18. 18.
Chen B, Yau S-T and Yeh Y-N, Graph homotopy and Graham homotopy, Discrete Math. 241 (2001) 153–170
19. 19.
Chepoi V and Estellon B, Packing and covering $$\delta$$-hyperbolic spaces by balls, APPROX-RANDOM (2007) pp. 59–73Google Scholar
20. 20.
Eppstein D, Squarepants in a tree: sum of subtree clustering and hyperbolic pants decomposition, SODA ’2007, Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms (2007) pp. 29–38Google Scholar
21. 21.
Gavoille C and Ly O, Distance Labeling in Hyperbolic Graphs, in: Lecture Notes in Computer Science, vol. 3827 (2005) pp. 1071–1079
22. 22.
Ghys E and de la Harpe P, Sur les Groupes Hyperboliques d’après Mikhael Gromov, Progress in Mathematics 83, (1990) (Boston, MA: Birkhäuser Boston Inc.)
23. 23.
Gromov M, Hyperbolic groups, in: Essays in group theory edited by S M Gersten, M.S.R.I. Publ. 8 (1987) (Springer) pp. 75–263Google Scholar
24. 24.
Hästö P A, Gromov hyperbolicity of the $$j_G$$ and $${\tilde{\jmath }}_G$$ metrics, Proc. Amer. Math. Soc. 134 (2006) 1137–1142
25. 25.
Hästö P A, Lindén H, Portilla A, Rodríguez J M and Tourís E, Gromov hyperbolicity of Denjoy domains with hyperbolic and quasihyperbolic metrics, J. Math. Soc. Jpn. 64 (2012) 247–261
26. 26.
Hästö P A, Portilla A, Rodríguez J M and Tourís E, Gromov hyperbolic equivalence of the hyperbolic and quasihyperbolic metrics in Denjoy domains, Bull. Lond. Math. Soc. 42 (2010) 282–294
27. 27.
Imrich W and Klavžar S, Product Graphs: Structure and Recognition, Wiley Series in Discrete Mathematics and Optimization (2000)Google Scholar
28. 28.
Jonckheere E and Lohsoonthorn P, A hyperbolic geometry approach to multipath routing, Proceedings of the 10th Mediterranean Conference on Control and Automation (MED 2002), Lisbon, Portugal, July 2002, FA5-1 (2002)Google Scholar
29. 29.
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 (2002) 45–60Google Scholar
30. 30.
Jonckheere E A and Lohsoonthorn P, Geometry of network security, Amer. Control Conf. ACC (2004) 111–151Google Scholar
31. 31.
Karlsson A and Noskov G A, The Hilbert metric and Gromov hyperbolicity, Enseign. Math. 48 (2002) 73–89
32. 32.
Kraner-Šumenjaka T, Pavlicb P and Tepeh A, On the Roman domination in the lexicographic product of graphs, Discrete Appl. Math. 160(13–14) (2012) 2030–2036
33. 33.
Krauthgamer R and Lee J R, Algorithms on negatively curved spaces, FOCS 2006 (2006)Google Scholar
34. 34.
Krioukov D, Papadopoulos F, Kitsak M, Vahdat A and Boguñá M, Hyperbolic geometry of complex networks, Phys. Rev. E 82 (2010) 036106
35. 35.
Lindén H, Gromov hyperbolicity of certain conformal invariant metrics, Ann. Acad. Sci. Fenn. Math. 32(1) (2007) 279–288
36. 36.
Michel J, Rodríguez J M, Sigarreta J M and Villeta M, Gromov hyperbolicity in cartesian product graphs, Proc. Indian Acad. Sci. (Math. Sci.) 120 (2010) 1–17
37. 37.
Michel J, Rodríguez J M, Sigarreta J M and Villeta M, Hyperbolicity and parameters of graphs, Ars Comb. 100 (2011) 43–63
38. 38.
Narayan O and Saniee I, Large-scale curvature of networks, Phys. Rev. E 84 (2011) 066108
39. 39.
Oshika K, Discrete groups (2002) (AMS Bookstore)Google Scholar
40. 40.
Pestana D, Rodríguez J M, Sigarreta J M and Villeta M, Gromov hyperbolic cubic graphs, Central Eur. J. Math. 10(3) (2012) 1141–1151
41. 41.
Portilla A, Rodríguez J M, Sigarreta J M and Vilaire J-M, Gromov hyperbolic tessellation graphs, Utilitas Math. 97 (2015) 193–212
42. 42.
Portilla A, Rodríguez J M and Tourís E, Gromov hyperbolicity through decomposition of metric spaces II, J. Geom. Anal. 14 (2004) 123–149
43. 43.
Portilla A, Rodríguez J M and Tourís E, The topology of balls and Gromov hyperbolicity of Riemann surfaces, Diff. Geom. Appl. 21 (2004) 317–335
44. 44.
Portilla A, Rodríguez J M and Tourís E, Stability of Gromov hyperbolicity, J. Advan. Math Stud. 2 (2009) 1–20
45. 45.
Power S C, Infinite lexicographic products of triangular algebras, Bull. Lond. Math. Soc. 27 (1995) 273–277
46. 46.
Rodríguez J M, Characterization of Gromov hyperbolic short graphs, Act. Math. Sin. 30 (2014) 197–212
47. 47.
Rodríguez J M, Sigarreta J M, Vilaire J-M and Villeta M, On the hyperbolicity constant in graphs, Discrete Math. 311 (2011) 211–219
48. 48.
Rodríguez J M and Tourís E, Gromov hyperbolicity through decomposition of metric spaces, Acta Math. Hung. 103 (2004) 53–84
49. 49.
Rodríguez J M and Tourís E, A new characterization of Gromov hyperbolicity for negatively curved surfaces, Publ. Mat. 50 (2006) 249–278
50. 50.
Rodríguez J M and Tourís E, Gromov hyperbolicity of Riemann surfaces, Acta Math. Sin. 23 (2007) 209–228
51. 51.
Saputro S W, Simanjuntak R, Uttunggadewa S, Assiyatun H, Baskoro E T, Salman A N M and Bača M, The metric dimension of the lexicographic product of graphs, Discrete Math. 313(9) (2013) 1045–1051
52. 52.
Shavitt Y, Tankel T, On internet embedding in hyperbolic spaces for overlay construction and distance estimation, INFOCOM 2004 (2004)Google Scholar
53. 53.
Sigarreta J M, Hyperbolicity in median graphs, Proc. Indian Acad. Sci. (Math. Sci.) 123 (2013) 455–467
54. 54.
Tourís E, Graphs and Gromov hyperbolicity of non-constant negatively curved surfaces, J. Math. Anal. Appl. 380 (2011) 865–881
55. 55.
Verbeek K and Suri S, Metric embeddings, hyperbolic space and social networks, in: Proceedings of the 30th Annual Symposium on Computational Geometry (2014) pp. 501–510Google Scholar
56. 56.
Wu Y, Zhang C, Chordality and hyperbolicity of a graph, Electr. J. Comb. 18 (2011) P43
57. 57.
Yang C and Xu J, Connectivity of lexicographic product and direct product of graphs, Ars Comb. 111 (2013) 3–12
58. 58.
Zhang X, Liu J and Meng J, Domination in lexicographic product graphs, Ars Comb. 101 (2011) 251–256

## Authors and Affiliations

• Walter Carballosa
• 1
• 2
• Amauris de la Cruz
• 3
• José M Rodríguez
• 3
1. 1.Department of Mathematics and StatisticsFlorida International UniversityMiamiUSA
2. 2.Department of MathematicsMiami Dade CollegeMiamiUSA