Let G be a connected graph, and let d(a, b) denotes the shortest path distance between vertices a and b of G. The graph G is δ-hyperbolic if for any vertices a, b, c, d of G, the two largest of the three sums S1 = d(a,b) + d(c,d), S2 = d(a,c) + d(b,d), and S3 = d(a,d) + d(b,c) differ by at most 2δ. This can be determined in time O(n^4) which could be prohibitive for large graphs. In this document, we propose an exact algorithm for determining the hyperbolicity of a graph that is scalable for large graphs. The time complexity of this algorithm is a function of the size of the largest bi-connected component of the graph, of the shortest path distance distribution in this component and of the value of the hyperbolicity. In the worst case, the time complexity remains in O(n^4), but it is much faster in practice. Indeed, it allowed us to compute the exact hyperbolicity of all maps of the autonomous systems of the Internet provided by CAIDA and DIMES. We also propose both a multiplicative factor and an additive constant approximation algorithms. Finally, we also analyze further the time complexity of our exact algorithm for several class of graphs.
