Graphs and Combinatorics

, Volume 31, Issue 5, pp 1311–1324 | Cite as

Planarity and Hyperbolicity in Graphs

  • Walter Carballosa
  • Ana Portilla
  • José M. Rodríguez
  • José M. Sigarreta
Original Paper


If X is a geodesic metric space and \(x_1,x_2,x_3\) are three points in \(X\), a geodesic triangle \(T=\{x_1,x_2,x_3\}\) is the union of 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 \(X\). 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. In this paper we obtain criteria which allow us to decide, for a large class of graphs, whether they are hyperbolic or not. We are especially interested in the planar graphs which are the “boundary” (the \(1\)-skeleton) of a tessellation of the Euclidean plane. Furthermore, we prove that a graph obtained as the \(1\)-skeleton of a general CW \(2\)-complex is hyperbolic if and only if its dual graph is hyperbolic.


Tessellation Planar graph Gromov hyperbolicity  CW complex Dual graph 

Mathematics Subject Classification

05C69 05A20 05C50 



This work was partly supported by the Spanish Ministry of Science and Innovation through projects MTM 2009-07800, MTM 2009-12740-C03-01 and MTM 2008-02829-E.


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Copyright information

© Springer Japan 2014

Authors and Affiliations

  • Walter Carballosa
    • 1
  • Ana Portilla
    • 2
  • José M. Rodríguez
    • 3
  • José M. Sigarreta
    • 1
  1. 1.Faculty of MathematicsAutonomous University of GuerreroAcapulcoMexico
  2. 2.St. Louis University (Madrid Campus)MadridSpain
  3. 3.Department of MathematicsCarlos III University of MadridLeganésSpain

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