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# assgt8 - w i are capable of performing jobs t j as shown in...

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MATH 239 Fall 2008 Assignment 8 Due Friday, November 28, noon. 1. The purpose of this problem is to prove that every non-maximum matching admits an aug- menting path. Let M be a matching in a graph G . Let N be a matching which is larger than M . Let H be the subgraph of G whose edges are ( M N ) \ ( M N ) and whose vertices are the set of vertices of G . (a) Show that every component of H consists of either a path or a cycle. (b) Prove that every path in H is an alternating path of G with respect to M . (c) Prove that at least one path in H is an augmenting path of G with respect to M . 2. Suppose that M is a matching of G which is not contained in any larger matching, and that M 0 is a maximum matching of G . Prove that | M 0 | ≤ 2 | M | . 3. (a) Let G be a bipartite graph, and let d be an integer such that every vertex of G has degree at most d . Prove that G has a matching of size at least q/d . (b) Give an example of a graph G and an integer d such that every vertex of G has degree at most d , but G has no matching of size at least q/d . 4. Workers
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Unformatted text preview: w i are capable of performing jobs t j as shown in the following table. Worker capable of these jobs w 1 t 1 , t 2 , t 3 w 2 t 1 , t 2 , t 5 w 3 t 2 , t 4 , t 6 w 4 t 3 , t 4 , t 5 w 5 t 4 , t 6 w 6 t 2 , t 6 (a) Draw the bipartite graph G whose vertices are the the w i and t j and whose edges represent which workers are capable of which jobs. (b) Starting with the matching M = {{ w 1 ,t 1 } , { w 2 ,t 2 } , { w 3 ,t 4 } , { w 4 ,t 5 } , { w 6 ,t 6 }} use the bipartite matching algorithm to ﬁnd a maximum matching for G . (c) Does G have a perfect matching? (d) Worker w 2 becomes partially incapacitated and can no longer perform tasks t 1 or t 5 . Let H be the resulting bipartite graph. Starting with the matching M given above, use the bipartite matching algorithm to ﬁnd a minimum covering and a maximum matching in H . (e) Does H have a perfect matching?...
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