Flow trees for vertex-capacitated networks

Given a graph G = (V, E) with a cost function c (S) ≥ 0 ∀ S ⊆ V, we want to represent all possible min-cut values between pairs of vertices i and j. We consider also the special case with an additive cost c where there are vertex capacities c (v) ≥ 0∀ v ∈ V, and for a subset S ⊆ V, c (S) = ∑_{v ∈ S} c (v). We consider two variants of cuts: in the first one, separation, { i } and { j } are feasible cuts that disconnect i and j. In the second variant, vertex-cut, a cut-set that disconnects i from j does not include i or j. We consider both variants for undirected and directed graphs. We prove that there is a flow-tree for separations in undirected graphs. We also show that a compact representation does not exist for vertex-cuts in undirected graphs, even with additive costs. For directed graphs, a compact representation of the cut-values does not exist even with additive costs, for neither the separation nor the vertex-cut cases.