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# h2 - S are adjacent For example c,h,u is an independent set...

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MATH 4171 Graph Theory SPRING 2006 Homework Set II (5 problems) Due date : Monday 2-13-06 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 2. Prove that every closed walk of odd length must contain an odd cycle. 3. Prove that trees are precisely maximal acyclic graphs. That is, prove that the following are equivalent for any graph G : (1) G is a tree; (2) G is acyclic and for any two distinct vertices x and y of G , adding to G a new edge e between x and y always results in a graph that has at least one cycle. 4. Find a minimum cost spanning tree in the weighted graph above. Show your work. 5. Let G = ( V,E ) be a simple graph. A set S V is independent if no two vertices in
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Unformatted text preview: S are adjacent. For example, { c,h,u } is an independent set in the above graph. Suppose there is a weight w ( x ) (a nonnegative number) on each vertex x o± G . The maximum-independent-set-problem (MISP) is to fnd an independent set S such that w ( S ) := ∑ x ∈ S w ( x ) is maximized. The greedy algorithm ±or solving MISP works as ±ollows: We start with S = ∅ and at each iteration we add to S the next vertex x o± the largest weight ±or which S ∪ { x } is independent. Question: Does this greedy algorithm always produce an independent set o± the largest weight? You need to justi±y your conclusion. Important —– No Late Homework Will be Accepted —–...
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