hw7(2) - G also contain k edge-disjoint paths from v to u ?...

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CS 473 Homework 7 (due April 14, 2009) Spring 2009 CS 473: Undergraduate Algorithms, Spring 2009 Homework 7 Due Tuesday, April 14, 2009 at 11:59:59pm. Groups of up to three students may submit a single, common solution for this and all future homeworks. Please clearly write every group member’s name and NetID on every page of your submission. 1. A graph is bipartite if its vertices can be colored black or white such that every edge joins vertices of two different colors. A graph is d-regular if every vertex has degree d . A matching in a graph is a subset of the edges with no common endpoints; a matching is perfect if it touches every vertex. (a) Prove that every regular bipartite graph contains a perfect matching. (b) Prove that every d -regular bipartite graph is the union of d perfect matchings. 2. Let G = ( V , E ) be a directed graph where for each vertex v , the in-degree of v and out-degree of v are equal. Let u and v be two vertices G , and suppose G contains k edge-disjoint paths from u to v . Under these conditions, must
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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.

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