The graphs with the max-Mader-flow-min-multiway-cut property

We are given a graph G, an independant set S ⊂ V (G) of terminals, and a function w : V (G) → N. We want to know if the maximum w-packing of vertex-disjoint paths with extremities in S is equal to the minimum weight of a vertex-cut separating S. We call Mader-Mengerian the graphs with this property for each independant set S and each weight function w. We give a characterization of these graphs in term of forbidden minors, as well as a recognition algorithm and a simple algorithm to find maximum packing of paths and minimum multicuts in those graphs.