Planarity and Hyperbolicity in Graphs

If X is a geodesic metric space and $$x_1,x_2,x_3$$x1,x2,x3 are three points in $$X$$X, a geodesic triangle$$T=\{x_1,x_2,x_3\}$$T={x1,x2,x3} is the union of three geodesics $$[x_1x_2]$$[x1x2], $$[x_2x_3]$$[x2x3] and $$[x_3x_1]$$[x3x1] in $$X$$X. The space $$X$$X is $$\delta $$δ-hyperbolic$$($$(in the Gromov sense$$)$$) if any side of $$T$$T is contained in a $$\delta $$δ-neighborhood of the union of the two other sides, for every geodesic triangle $$T$$T in $$X$$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 $$1$$1-skeleton) of a tessellation of the Euclidean plane. Furthermore, we prove that a graph obtained as the $$1$$1-skeleton of a general CW $$2$$2-complex is hyperbolic if and only if its dual graph is hyperbolic.