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Unformatted text preview: MATH 4171 Graph Theory SPRING 2006 Homework Set III Solutions 1. Consider the two arcdisjoint 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 arcdisjoint s t dipaths; (c) Is there an augmenting path with respect to the three paths you obtained in (b)? If so, find it; if no, prove it. a b c d e f g s t h a b c d e f g s t h Solution. (a) sdfet is an augmenting path. (b) sabhgt , sdft , and set are these three paths. (c) No there are no more augmenting paths. This is justified by the s t cut E + ( X,Y ), where X = { s,a,b,c,d } , Y = { e,f,g,h,t } , and E + ( X,Y ) = { bh,df,se } , which has size three. 2. Consider the following network, where the number on each arc is the corresponding capacity. Let f be a flow with f ( sa ) = f ( ab ) = f ( bt ) = f ( sc ) = f ( cd ) = f ( dt ) = 1. Find a flow of the maximum value by repeatedly finding augmenting paths. Justify the maximality of your flow 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 Solution. sbadct is an augmenting path. Its forward arcs are sb , ad , and ct , while its backward arcs are ab and cd . The possible increasing amount on the forward arcs are 1, 2, and 3, respectively; the possible decreasing amount on the backward arcs are 1 and 1, respectively. So we can augment thepossible decreasing amount on the backward arcs are 1 and 1, respectively....
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 Spring '08
 Lax,R
 Graph Theory, Bipartite graph, Menger Theorem

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