h2 - S are adjacent. For example, { c,h,u } is an...

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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, fnd the distance ±rom 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 o± odd length must contain an odd cycle. 3. Prove that trees are precisely maximal acyclic graphs. That is, prove that the ±ollowing are equivalent ±or any graph G : (1) G is a tree; (2) G is acyclic and ±or any two distinct vertices x and y 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 i± 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 justiy your conclusion. Important No Late Homework Will be Accepted...
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This note was uploaded on 02/17/2009 for the course MATH 4171 taught by Professor Lax,r during the Spring '08 term at LSU.

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