For an undirected graph G = (V; E), the edge connectivity values between every pair of nodes of G can be recorded in a flow-equivalent tree that contains the edge connectivity value for a linear number of pairs of nodes. We generalize this result to show how we can efficiently recover a maximum set of disjoint paths between any pair of nodes of G by storing such sets for a linear number of pairs of nodes. At the heart of our result is an observation that combining two flow solutions of the same value, one between s and r and the second between r and t, into a feasible flow solution of value f between s and t, is equivalent to solving a stable matching problem on a bipartite multigraph. 1 Introduction Given an undirected graph G = (V; E) with jV j = n, let (s; t) be the st-edge connectivity of G, i.e., the maximum number of edge-disjoint st-paths. Gomory and Hu  showed that the edge connectivity function = f(s; t) : s; t 2 V g has a compact tree representation, i.e., there..