Unformatted text preview: G also contain k edgedisjoint paths from v to u ? Give a proof or a counterexample with explanation. 3. A ﬂow f is called acyclic if the subgraph of directed edges with positive ﬂow contains no directed cycles. A ﬂow is positive if its value is greater than 0. (a) A path ﬂow assigns positive values only to the edges of one simple directed path from s to t . Prove that every positive acyclic ﬂow can be written as the sum of a ﬁnite number of path ﬂows. (b) Describe a ﬂow in a directed graph that cannot be written as the sum of path ﬂows. (c) A cycle ﬂow assigns positive values only to the edges of one simple directed cycle. Prove that every ﬂow can be written as the sum of a ﬁnite number of path ﬂows and cycle ﬂows. (d) Prove that for any ﬂow f , there is an acyclic ﬂow with the same value as f . (In particular, this implies that some maximum ﬂow is acyclic.) 1...
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 Fall '08
 Chekuri,C
 Algorithms, Graph Theory, Glossary of graph theory, Bipartite graph, regular bipartite graph

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