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Unformatted text preview: MATH 4171 Graph Theory SPRING 2006 Homework Set II Solutions 1. For the graph below, find the distance from u to h , and a shortest u h path (with respect to the given numbers on each edge). Show your work. 1 1 2 2 3 3 4 5 7 a b c d e f g h u 6 2 7 5 9 1 1 1 2 5 5 1 1 2 2 3 3 4 5 7 a b c d e f g h u 6 2 7 5 9 1 1 1 2 5 5 Figure 1. A shortest u h path and minimum cost spanning tree. A shortest path is illustrated in the graph. We start with S = { u } . Then we have S = { u,a } and t ( a ) = 1. Then S = { u,a,b } with t ( b ) = 3. Then S = { u,a,b,d } with t ( d ) = 4. Then S = { u,a,b,d,e } with t ( e ) = 5. Then S = { u,a,b,d,e,g } with t ( g ) = 6. Then S = { u,a,b,d,e,g,f } with t ( f ) = 6. Then we have S = { u,a,b,d,e,g,f,c,h } with t ( h ) = t ( c ) = 7. So the distance from u to h is 7. 4. Find a minimum cost spanning tree in the weighted graph above. Show your work. We use the greedy algorithm, by finding a maximal acyclic subgraph H of the given graph. Let F be the set of edges of H . We started with F = . The smallest cost is 1 and there are five edges of that cost. Add these edges to F , one by one, and that does not create any cycles. At this point, F = { ua,eg,gc,cf,fh } ....
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 Spring '08
 Lax,R
 Graph Theory

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