Network flows: theory, algorithms, and applications
Network flows: theory, algorithms, and applications
Counterexamples for Directed and Node Capacitated Cut-Trees
SIAM Journal on Computing
Computing Maximum Flows in Undirected Planar Networks with Both Edge and Vertex Capacities
COCOON '08 Proceedings of the 14th annual international conference on Computing and Combinatorics
| Hi-index | 0.89 |
Given an edge-capacitated undirected graph G = (V, E, C) with edge capacity c:E ↦ R+, n = |V|, an s-t edge cut C of G is a minimal subset of edges whose removal from G will separate s from t in the resulting graph, and the capacity sum of the edges in C is the cut value of C. A minimum s-t edge cut is an s-t edge cut with the minimum cut value among all s-t edge cuts. A theorem given by Gomory and Hu states that there are only n-1 distinct values among the n(n-1)/2 minimum edge cuts in an edge-capacitated undirected graph G, and these distinct cuts can be compactly represented by a tree with the same node set as G, which is referred to the flow equivalent tree. In this paper we generalize their result to the node-edge cuts in a node-edge-capacitated undirected planar graph. We show that there is a flow equivalent tree for node-edge-capacitated undirected planar graphs, which represents the minimum node-edge cut for any pair of nodes in the graph through a novel transformation.