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Unformatted text preview: G also contain k edge-disjoint paths from v to u ? Give a proof or a counterexample with explanation. 3. A ow f is called acyclic if the subgraph of directed edges with positive ow contains no directed cycles. A ow is positive if its value is greater than 0. (a) A path ow assigns positive values only to the edges of one simple directed path from s to t . Prove that every positive acyclic ow can be written as the sum of a nite number of path ows. (b) Describe a ow in a directed graph that cannot be written as the sum of path ows. (c) A cycle ow assigns positive values only to the edges of one simple directed cycle. Prove that every ow can be written as the sum of a nite number of path ows and cycle ows. (d) Prove that for any ow f , there is an acyclic ow with the same value as f . (In particular, this implies that some maximum ow is acyclic.) 1...
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This note was uploaded on 01/22/2012 for the course CS 573 taught by Professor Chekuri,c during the Fall '08 term at University of Illinois, Urbana Champaign.
- Fall '08