assign-2-2018.pdf

# 3 let g be the bipartite graph below and let m v 1 w

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3. Let G be the bipartite graph below and let M = { v 1 w 1 , v 3 w 3 , v 4 w 4 } be the matching in G shown with bold edges. v 1 v 2 v 3 v 4 v 5 w 1 w 2 w 3 w 4 w 5 w 6 Figure 3: Question 3 (a) Grow a maximal alternating tree in G with respect to M with root v 2 . (b) Use this tree to obtain a matching in G with four edges. (c) Is it possible to find an augmenting path with respect to this new matching? If so give such a path and augment along it to obtain a matching with five edges. Otherwise give reasons why such an augmenting path does not exist. 4. Let M be a matching in a graph G , and let S be the set of vertices matched by M . Prove that there exists a maximum matching in G under which all vertices in S are matched. 5. Let N be the following network with source x and sink y . The arc capacities are shown beside each arc, and in brackets next to each capacity is the current value of the flow in the arc. (a) Apply the labelling algorithm to find a maximum flow in N starting with the flows as shown. Show each of the remaining stages of the algorithm, and state the value of the maximum flow. (b) Using your results in (a), find a minimum cut in N . 2

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a e d 6 (4) b x y c 3 (3) 4 (3) 3 (1) 4 (1) 2 (1) 4 (0) 1 (1) 3 (0) 5 (0) 2 (1) Figure 4: Question 5 3
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