Graphs with small hyperbolicity constant

If X is a geodesic metric space and x1,x2,x3∈X, a geodesic triangle T={x1,x2,x3} is the union of the three geodesics [x1x2],[x2x3] and [x3x1] 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.