# Generalized Chordality, Vertex Separators and Hyperbolicity on Graphs

## Abstract

**:**

## 1. Introduction

## 2. Generalized Chordality and Minimal Vertex Separators

**Remark**

**1.**

**Definition**

**1.**

**Definition**

**2.**

**Remark**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Remark**

**3.**

**Proposition**

**1.**

**Proof.**

**Example**

**1.**

**Remark**

**4.**

**Definition**

**5.**

**Lemma**

**1.**

**Proof.**

**Remark**

**5.**

**Lemma**

**2.**

**Proof.**

**Definition**

**6.**

**Definition**

**7.**

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Corollary**

**1.**

**Corollary**

**2.**

## 3. Bottleneck Property

**Definition**

**8.**

**Remark**

**6.**

**Definition**

**9.**

**Proposition**

**2.**

**Proof.**

**Theorem**

**4.**

**Theorem**

**5.**

**Proof.**

**Corollary**

**3.**

**Definition**

**10.**

**Lemma**

**3.**

**Theorem**

**6.**

**Proof.**

**Remark**

**7.**

**Theorem**

**7.**

**Proof.**

**Example**

**2.**

**Remark**

**8.**

**Theorem**

**8.**

**Proof.**

**Corollary**

**4.**

**Remark**

**9.**

**Remark**

**10.**

**Remark**

**11.**

## 4. Minimal Vertex r-Separators

**Definition**

**11.**

**Remark**

**12.**

**Definition**

**12.**

**Remark**

**13.**

**Lemma**

**4.**

**Proof.**

**Theorem**

**9.**

**Proof.**

**Theorem**

**10.**

**Proof.**

**Corollary**

**5.**

**Corollary**

**6.**

**Corollary**

**7.**

## 5. Neighbor Separators

**Definition**

**13.**

**Theorem**

**11.**

**Proof.**

**Corollary**

**8.**

**Proposition**

**3.**

**Proof.**

**Definition**

**14.**

**Proposition**

**4.**

**Proof.**

**Proposition**

**5.**

**Proof.**

**Corollary**

**9.**

## 6. Neighbor Obstructors

**Definition**

**15.**

**Definition**

**16.**

**Theorem**

**12.**

**Proof.**

**Definition**

**17.**

**Proposition**

**6.**

**Proof.**

**Proposition**

**7.**

**Proof.**

**Theorem**

**13.**

**Proof.**

**Example**

**3.**

**Remark**

**14.**

**Remark**

**15.**

**Proposition**

**8.**

**Proof.**

## Acknowledgments

## Conflicts of Interest

## References

- Gromov, M. Hyperbolic groups. In Essays in Group Theory; Gersten, S.M., Ed.; Mathematical Science Research Institute Publications; Springer: New York, NY, USA, 1987; Volume 8, pp. 75–263. [Google Scholar]
- Bridson, M.; Haefliger, A. Metric Spaces of Non-Positive Curvature; Springer: Berlin, Germany, 1999. [Google Scholar]
- Burago, D.; Burago, Y.; Ivanov, S. A course in metric geometry. In Graduate Studies in Mathematics; AMS: Providence, RI, USA, 2001; Volume 33. [Google Scholar]
- Buyalo, S.; Schroeder, V. Elements of Asymptotic Geometry. In EMS Monographs in Mathematics; European Mathematical Society: Zürich, Switzerland, 2007. [Google Scholar]
- Gyhs, E.; de la Harpe, P. Sur le groupes hyperboliques d’après Mikhael Gromov. In Progress in Math; Birkhäuser: Boston, MA, USA, 1990; Volume 83. [Google Scholar]
- Väisälä, J. Gromov hyperbolic spaces. Expos. Math.
**2005**, 23, 187–231. [Google Scholar] [CrossRef] - Bermudo, S.; Rodríguez, J.M.; Rosario, O.; Sigarreta, J.M. Small values of the hyperbolicity constant in graphs. Discret. Math.
**2016**, 339, 3073–3084. [Google Scholar] [CrossRef] - Bermudo, S.; Rodríguez, J.M.; Sigarreta, J.M. Computing the hyperbolicity constant. Comput. Math. Appl.
**2011**, 62, 4592–4595. [Google Scholar] [CrossRef] - Carballosa, W.; Pestana, D.; Rodríguez, J.M.; Sigarreta, J.M. Distortion of the hyperbolicity constant of a graph. Electron. J. Comb.
**2012**, 19, # P67. [Google Scholar] [PubMed] - Carballosa, W.; Rodríguez, J.M.; Sigarreta, J.M.; Villeta, M. Gromov hyperbolicity of line graphs. Electron. J. Comb.
**2011**, 18, # P210. [Google Scholar] - Bermudo, S.; Rodríguez, J.M.; Sigarreta, J.M.; Vilaire, J.-M. Gromov hyperbolic graphs. Discret. Math.
**2013**, 313, 1575–1585. [Google Scholar] [CrossRef] - Chepoi, V.; Dragan, F.F.; Estellon, B.; Habib, M.; Vaxes, Y. Notes on diameters, centers, and approximating trees of δ-hyperbolic geodesic spaces and graphs. Electron. Notes Discret. Math.
**2008**, 31, 231–234. [Google Scholar] [CrossRef] - Frigerio, R.; Sisto, A. Characterizing hyperbolic spaces and real trees. Geom. Dedicata
**2009**, 142, 139–149. [Google Scholar] [CrossRef] - Hästö, P.A. Gromov hyperbolicity of the j
_{G}and ${\tilde{\u0237}}_{G}$ metrics. Proc. Am. Math. Soc.**2006**, 134, 1137–1142. [Google Scholar] [CrossRef] - Michel, J.; Rodríguez, J.M.; Sigarreta, J.M.; Villeta, M. Hyperbolicity and parameters of graphs. Ars Comb.
**2011**, 100, 43–63. [Google Scholar] - Pestana, D.; Rodríguez, J.M.; Sigarreta, J.M.; Villeta, M. Gromov hyperbolic cubic graphs. Cent. Eur. J. Math.
**2012**, 10, 1141–1151. [Google Scholar] [CrossRef] - Portilla, A.; Rodríguez, J.M.; Sigarreta, J.M.; Vilaire, J.-M. Gromov hyperbolic tessellation graphs. Util. Math.
**2015**, 97, 193–212. [Google Scholar] - Portilla, A.; Rodríguez, J.M.; Tourís, E. Gromov hyperbolicity through decomposition of metric spaces II. J. Geom. Anal.
**2004**, 14, 123–149. [Google Scholar] [CrossRef] - Portilla, A.; Rodríguez, J.M.; Tourís, E. Stability of Gromov hyperbolicity. J. Adv. Math. Stud.
**2009**, 2, 77–96. [Google Scholar] - Portilla, A.; Tourís, E. A characterization of Gromov hyperbolicity of surfaces with variable negative curvature. Publ. Matorsz.
**2009**, 53, 83–110. [Google Scholar] [CrossRef][Green Version] - Rodríguez, J.M.; Sigarreta, J.M. Bounds on Gromov hyperbolicity constant in graphs. Proc. Indian Acad. Sci. Math. Sci.
**2012**, 122, 53–65. [Google Scholar] [CrossRef] - Rodríguez, J.M.; Sigarreta, J.M.; Torres-Nuñez, Y. Computing the hyperbolicity constant of a cubic graph. Int. J. Comput. Math.
**2014**, 91, 1897–1910. [Google Scholar] [CrossRef] - Rodríguez, J.M.; Sigarreta, J.M.; Vilaire, J.-M.; Villeta, M. On the hyperbolicity constant in graphs. Discret. Math.
**2011**, 311, 211–219. [Google Scholar] [CrossRef] - Sigarreta, J.M. Hyperbolicity in median graphs. Proc. Indian Acad. Sci. Math. Sci.
**2013**, 123, 455–467. [Google Scholar] [CrossRef] - Tourís, E. Graphs and Gromov hyperbolicity of non-constant negatively curved surfaces. J. Math. Anal. Appl.
**2011**, 380, 865–881. [Google Scholar] [CrossRef] - Dress, A.; Holland, B.; Huber, K.T.; Koolen, J.H.; Moulton, V.; Weyer-Menkhoff, J. Δ additive and Δ ultra-additive maps, Gromov’s trees, and the Farris transform. Discret. Appl. Math.
**2005**, 146, 51–73. [Google Scholar] [CrossRef] - Dress, A.; Moulton, V.; Terhalle, W. T-theory: An overview. Eur. J. Comb.
**1996**, 17, 161–175. [Google Scholar] [CrossRef] - Clauset, A.; Moore, C.; Newman, M.E.J. Hierarchical structure and the prediction of missing links in networks. Nature
**2008**, 453, 98–101. [Google Scholar] [CrossRef] [PubMed] - Krioukov, D.; Papadopoulos, F.; Kitsak, M.; Vahdat, A.; Boguñá, M. Hyperbolic geometry of complex networks. Phys. Rev. E
**2010**, 82, 036106. [Google Scholar] [CrossRef] [PubMed] - Shang, Y. Lack of Gromov-hyperbolicity in small-world networks. Cent. Eur. J. Math.
**2012**, 10, 1152–1158. [Google Scholar] [CrossRef] - Shang, Y. Non-hyperbolicity of random graphs with given expected degrees. Stoch. Model.
**2013**, 29, 451–462. [Google Scholar] [CrossRef] - 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.
**2002**, 8, 45–60. [Google Scholar] - Jonckheere, E.A.; Lohsoonthorn, P. Geometry of network security. Proc. Am. Control Conf.
**2004**, 2, 976–981. [Google Scholar] - Jonckheere, E.A.; Lou, M.; Bonahon, F.; Baryshnikov, Y. Euclidean versus hyperbolic congestion in idealized versus experimental networks. Int. Math.
**2011**, 7, 1–27. [Google Scholar] [CrossRef] - Sreenivasa Kumar, P.; Veni Madhavan, C.E. Minimal vertex separators of chordal graphs. Discret. Appl. Math.
**1998**, 89, 155–168. [Google Scholar] [CrossRef] - Blair, J.; Peyton, B. An introduction to chordal graphs and clique trees, Graph Theory and Sparse Matrix Multiplication. In IMA Volumes in Mathematics and its Applications; Springer: Berlin, Germany, 1993; Volume 56, pp. 1–29. [Google Scholar]
- Brinkmann, G.; Koolen, J.; Moulton, V. On the hyperbolicity of chordal graphs. Ann. Comb.
**2001**, 5, 61–69. [Google Scholar] [CrossRef] - Wu, Y.; Zhang, C. Hyperbolicity and chordality of a graph. Electron. J. Comb.
**2011**, 18, # P43. [Google Scholar] - Bermudo, S.; Carballosa, W.; Rodríguez, J.M.; Sigarreta, J.M. On the hyperbolicity of edge-chordal and path-chordal graphs. Filomat
**2016**, 30, 2599–2607. [Google Scholar] [CrossRef] - Martínez-Pérez, A. Chordality properties and hyperbolicity on graphs. Electron. J. Comb.
**2016**, 23, # P3.51. [Google Scholar] - Dirac, G.A. On rigid circuit graphs. Abh. Math. Semin. Univ. Hambg.
**1961**, 25, 71–76. [Google Scholar] [CrossRef] - Krithika, R.; Mathew, R.; Narayanaswamy, N.S.; Sadagopan, N. A Dirac-type Characterization of k-chordal Graphs. Discret. Math.
**2013**, 313, 2865–2867. [Google Scholar] [CrossRef] - Anandkumar, A.; Tan, V.; Huang, F.; Willsky, A.S. High-dimensional Gaussian graphical model selection: Walk summability and local separation criterion. J. Mach. Learn. Res.
**2012**, 13, 2293–2337. [Google Scholar] - Manning, J.F. Geometry of pseudocharacters. Geom. Topol.
**2005**, 9, 1147–1185. [Google Scholar] [CrossRef] - Bestvina, M.; Bromberg, K.; Fujiwara, K. Constructing group actions on quasi-trees and applications to mapping class groups. Publ. Math. l’IHÉS
**2015**, 122, 1–64. [Google Scholar] [CrossRef] - Cashen, C.H. A Geometric Proof of the Structure Theorem for Cyclic Splittings of Free Groups. Topol. Proc.
**2017**, 50, 335–349. [Google Scholar] - Martínez-Pérez, A. Real-valued functions and metric spaces quasi-isometric to trees. Ann. Acad. Sci. Fenn. Math.
**2012**, 37, 525–538. [Google Scholar] [CrossRef] - Rodríguez, J.M.; Tourís, E. Gromov hyperbolicity through decomposition of metric spaces. Acta Math. Hung.
**2004**, 103, 53–84. [Google Scholar] [CrossRef] - Martínez-Pérez, A. Quasi-isometries between visual hyperbolic spaces. Manuscr. Math.
**2012**, 137, 195–213. [Google Scholar] [CrossRef] - Cao, J. Cheeger isoperimetric constants of Gromov-hyperbolic spaces with quasi-pole. Commun. Contemp. Math.
**2000**, 4, 511–533. [Google Scholar] [CrossRef] - Martínez-Pérez, A.; Rodríguez, J.M. Cheeger isoperimetric constant of Gromov hyperbolic manifolds and graphs. Commun. Contemp. Math.
**2017**, in press. [Google Scholar] [CrossRef] - Bieri, R.; Geoghegan, R. Limit sets for modules over groups on CAT(0) spaces: From the Euclidean to the hyperbolic. Proc. Lond. Math. Soc.
**2016**, 112, 1059–1102. [Google Scholar] [CrossRef] - Hughes, B. Trees and ultrametric spaces: A categorical equivalence. Adv. Math.
**2004**, 189, 148–191. [Google Scholar] [CrossRef] - Martínez-Pérez, A.; Morón, M.A. Uniformly continuous maps between ends of ℝ-trees. Math. Z.
**2009**, 263, 583–606. [Google Scholar] [CrossRef]

**Figure 2.**Every minimal vertex separator has diameter at most two, but the graph is not $(k,1)$-chordal for any $k>0$.

**Figure 3.**Satisfying the bottleneck property does not imply the existence of minimal vertex separators with uniformly-bounded diameters.

© 2017 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Martínez-Pérez, Á.
Generalized Chordality, Vertex Separators and Hyperbolicity on Graphs. *Symmetry* **2017**, *9*, 199.
https://doi.org/10.3390/sym9100199

**AMA Style**

Martínez-Pérez Á.
Generalized Chordality, Vertex Separators and Hyperbolicity on Graphs. *Symmetry*. 2017; 9(10):199.
https://doi.org/10.3390/sym9100199

**Chicago/Turabian Style**

Martínez-Pérez, Álvaro.
2017. "Generalized Chordality, Vertex Separators and Hyperbolicity on Graphs" *Symmetry* 9, no. 10: 199.
https://doi.org/10.3390/sym9100199