h2a - MATH 4171 Graph Theory SPRING 2006 Homework Set II...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 } ....
View Full Document

Page1 / 2

h2a - MATH 4171 Graph Theory SPRING 2006 Homework Set II...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online