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

Abstract

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.

Keywords

Tessellation Planar graph Gromov hyperbolicity  CW complex Dual graph 

Mathematics Subject Classification

05C69 05A20 05C50 

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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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