h3 - s-t cut has size at least k . 4. Let G = ( V, E ) be a...

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MATH 4171 Graph Theory SPRING 2006 Homework Set III Due date : Monday 3-6-06 1. Consider the two arc-disjoint s - t dipaths, sabhgt and seft , in the following digraph. (a) Find an augmenting path with respect to these two paths; (b) Using the given two paths and your augmenting path to get three arc-disjoint s - t dipaths; (c) Is there an augmenting path with respect to the three paths you obtained in (b)? If so, ±nd it; if no, prove it. a b c d e f g s t h 2. Consider the following network, where the number on each arc is the corresponding capacity. Let f be a ²ow with f ( sa ) = f ( ab ) = f ( bt ) = f ( sc ) = f ( cd ) = f ( dt ) = 1. Find a ²ow of the maximum value by repeatedly ±nding augmenting paths. Justify the maximality of your ²ow by exhibiting an s - t cut of minimum capacity. You need to show your work. s t a b c d 1 2 1 1 2 2 2 3 3 3. Prove the edge-version of Menger Theorem by using its arc-version. Menger Theorem (Edge-version). Let G be an undirected graph with two speci±ed vertices s and t . Then G has k edge-disjoint s - t paths if and only if every
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Unformatted text preview: s-t cut has size at least k . 4. Let G = ( V, E ) be a 9-connected graph and let A, B V with A B = and | A | = | B | = 9. Prove that G has 9 vertex-disjoint paths between A and B . 5. Let us consider the following greedy algorithm for the maximum matching problem. Suppose the input graph is G = ( V, E ) with E = { e 1 , e 2 , ..., e m } . We start with M = . What we do at iteration i (1 i m ) is: if M { e i } is a matching, let M := M { e } ; if M { e i } is not a matching, keep the current M unchanged. After m iterations, we output M . (1) Find a graph G for which the algorithm does not produce a maximum matching. (2) If the algorithm produces a matching of size 2006 from an input graph G , is it possible for G to have a matching of size 4171 or more? Important No Late Homework Will be Accepted...
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