Planarity and Hyperbolicity in Graphs

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.