The shortest-path metric d of a graph G = (V, E) is called δ-hyperbolic if for any four vertices u, v, w, x ∈ X the two larger of the three sums d(u, v) + d(w, x), d(u, w) + d(v, x), d(u, x) + d(v, w) differ by at most δ. In this paper, we characterize the graphs with 1-hyperbolic metrics in terms of a convexity condition and forbidden isometric subgraphs.