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Planarity and Hyperbolicity in Graphs


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.

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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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Correspondence to Walter Carballosa.

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Carballosa, W., Portilla, A., Rodríguez, J.M. et al. Planarity and Hyperbolicity in Graphs. Graphs and Combinatorics 31, 1311–1324 (2015).

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  • Tessellation
  • Planar graph
  • Gromov hyperbolicity
  • CW complex
  • Dual graph

Mathematics Subject Classification

  • 05C69
  • 05A20
  • 05C50