Flow trees for vertex-capacitated networks

Abstract

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.