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Unformatted text preview: CME305 Sample Midterm II 1. Matchings and Vertex Covers (a) Define what a matching in G is. (b) Define what a vertex cover of G is. (c) Let M be a maximum matching and C a minimum vertex cover. Show that  M   C  2  M  . Solution: 1. A matching M is a subset of E such that no two edges share an end point. 2. A vertex cover C is a subset of V such that all edges are incident to at least one element of C . 3. Let M be a maximum matching and C a minimum vertex cover. It is easy to see that we need at least  M  nodes to cover all the edges of M . Hence  M   C  . Now let us prove that V ( M ), the vertices in the matching M , is a vertex cover. Assume it isnt, then there is at least one edge e that is not covered by V ( M ). It is easy to see that this implies that M { e } is a matching so M is not maximum. Contradiction. Furthermore, we know that  V ( M )  = 2  M  , hence  C  2  M  ....
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This document was uploaded on 06/17/2010.
 Winter '09

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