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Unformatted text preview: Not to be turned in. Solutions will be posted Monday July 25th. Covering sections 7.2-7.4. Math 239 Assignment 11 1. Show that | X | = | X | + | Y | - | U | where U is the set of unsaturated vertices in the XY-construction. Solution: For two sets A,B such that A B , let B- A denote the set of elements in B that are not in A . First, every x X is unsaturated and every x X- X is saturated. Second, if y Y- U , it is joint to some x X- X by an edge in the matching. These two facts, together with the de ning property of a matching that the edges share no common ends, implies | X- X | = | Y- U | . Thus | X | = | X | + | X- X | = | X | + | Y- U | = | X | + | Y | - | U | . 2. Let G be a bipartite graph with bipartition A,B , and M be a matching of G . (a) Show that, for any D A , | M | | A | - | D | + | N ( D ) | . (b) Show that the size of a maximum matching in G is equal to min D A | A | - | D | + | N ( D ) | . Solution: (a) Since G is bipartite, each edge e i in the matching has one end a i A and one end b i B . By the property of a matching, these a i 's and b i 's are distinct. Now, given any D A , we can express | M | as a sum of the number of a i s in D , and the number of a i s in A- D . The former is equal to the number of b i 's in N ( D ) , and is thus at most | N ( D ) | . The latter is at most | A- D | . Together: | M | | N ( D ) | + | A- D | = | N ( D ) | + | A | - | D | . (b) We provide two solutions here. In the rst solution, we nd D that attains the minimum if M is maximum. Apply the XY-construction with respect to M . Then, there are no edges from X to B- Y (else there will be an augmenting path contradicting maximality of M ). If we choose D = X , then N...
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This note was uploaded on 08/13/2011 for the course MATH 239 taught by Professor M.pei during the Spring '09 term at Waterloo.
- Spring '09