Graphs with small hyperbolicity constant

Abstract If X is a geodesic metric space and x 1 , x 2 , x 3 ∈ X , a geodesic triangle T = { x 1 , x 2 , x 3 } is the union of the three geodesics [ x 1 x 2 ] , [ x 2 x 3 ] and [ x 3 x 1 ] in X. The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in a δ-neighborhood of the union of the two other sides, for every geodesic triangle T in X. We denote by δ ( X ) the sharpest hyperbolicity constant of X, i.e. δ ( X ) : = inf ⁡ { δ ≥ 0 : X is δ-hyperbolic}. In this paper we study the graphs with small hyperbolicity constant.

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