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.

• Type: Journal
• Year: 2007
• Where: DAM
• Authors: Refael Hassin, Asaf Levin
• Keywords: Additive Costs, Cost Function, DAM 2007, Possible Min-cut Values