On Odd Cuts and Plane Multicommodity Flows

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Let Τbe an even subset of the vertices of a graph G.A T-cut is an edge-cutset of the graph which divides Τinto two odd sets. We prove that if Gis bipartite, then the maximum number of disjoint T-cuts is equal to the minimum cardinality of a set of edges which meets every T-cut. (A weaker form of this was proved by Edmonds and Johnson.) We deduce a solution to the real-valued multi-commodity flow problem for a class of planar graphs; and we also solve the integral 2-commodity flow problem for the same class of graphs, by a further analysis of the T-cut problem when T = 4.

Original languageEnglish (US)
Pages (from-to)178-192
Number of pages15
JournalProceedings of the London Mathematical Society
Issue number1
StatePublished - Jan 1981
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • General Mathematics


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