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

Abstract

We are given a graph $G$, an independant set $\mathcal{S}\subset V(G)$ of \emph{terminals}, and a function $w:V(G)\to \mathbb{N}$. We want to know if the maximum $w$-packing of vertex-disjoint paths with extremities in $\mathcal{S}$ is equal to the minimum weight of a vertex-cut separating $\mathcal{S}$. We call \emph{Mader-Mengerian} the graphs with this property for each independant set $\mathcal{S}$ 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.