sol9 - MATH 239 Assignment 9 - Suggested Solutions 1. Here...

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Unformatted text preview: MATH 239 Assignment 9 - Suggested Solutions 1. Here we have | M | < | A | , and from the XY-construction, we will create a minimum cover C = Y A \ X with | C | = | M | < | A | . But then in the language of the proof of Halls Theorem (7.3.1), we have D = A \ C = A C = X , and thus | N ( X ) | < | X | , as shown in the proof of Halls 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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sol9 - MATH 239 Assignment 9 - Suggested Solutions 1. Here...

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