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Unformatted text preview: MATH 239 Assignment 9  Suggested Solutions 1. Here we have  M  <  A  , and from the XYconstruction, we will create a minimum cover C = Y ∪ A \ X with  C  =  M  <  A  . But then in the language of the proof of Hall’s Theorem (7.3.1), we have D = A \ C = A ∩ C = X , and thus  N ( X )  <  X  , as shown in the proof of Hall’s Theorem on page 190 of the Course Notes. 2. Applying the bipartite matching algorithm to G with the given bipartition A,B , and matching M consisting of the thick edges, we obtain X = { 3 , 5 } , and then construct the trees below, rooted at vertices 5, 3, to obtain X = A , Y = B . The set of vertices not 2 d b a 4 1 e c 3 5 saturated by M in Y is U = { a,e } , which is nonempty, and one augmenting path in the above trees is 5 e 4 d . Switching the edges in this path, we obtain the new matching M ′ , given by the thick edges below. For the matching M ′ , we have X = { 3 } , and then construct 1 2 3 4 5 a b c d e A B G the tree below, rooted at vertex 3, to obtain X = { 1 , 3 , 5 } , Y = { c,e } . Here we have....
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 Fall '09
 M.PEI
 Math, Empty set, Bipartite graph

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