To give an example of a graph with at least4 vertices for which set of maximum weight edges S G is equal to the maximum weight spanning tree T G of graph G.

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Answer 1

Question

Chapter 35, Problem 6P

a.

Program Plan Intro

To give an example of a graph with at least4 vertices for which set of maximum weight edges SG is equal to the maximum weight spanning tree TG of graph G.

b.

Program Plan Intro

To give an example of a graph with at least4 vertices for which set of maximum weight edges SG is not equal to the maximum weight spanning tree TG of graph G.

c.

Program Plan Intro

To prove that SG⊆TG for any graph G.

d.

Program Plan Intro

To prove that w(SG)⊆w(TG)/2 for any graph G.

e.

Program Plan Intro

To give an O(V+E) -time algorithm to compute a 2-approximation to the maximum spanning tree.

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Students have asked these similar questions

Let G = (V, E) be an undirected graph and each edge e ∈ E is associated with a positive weight ℓ(e).For simplicity we assume weights are distinct. Is the following statement true or false? Prove by contradiction or counterexample. Let T be a minimum spanning tree for the graph with the original weight. Suppose we replace eachedge weight ℓ(e) with ℓ(e)^2, then T is still a minimum spanning tree.

Let G = (V, E) be an undirected graph and each edge e ∈ E is associated with a positive weight ℓ(e).For simplicity we assume weights are distinct. Is the following statement true or false? Let T be a minimum spanning tree for the graph with the original weight. Suppose we replace eachedge weight ℓ(e) with ℓ(e)^2, then T is still a minimum spanning tree.

G = (V,E,W) is a weighted connected (undirected) graph where all edges have distinct weights except two edges e and e′ which have the same weight. Suppose there is a Minimum Spanning Tree of G containing both e and e′. Prove that G has a unique Minimum Spanning Tree.

Answer 2

Question

Chapter 35, Problem 6P

a.

Program Plan Intro

To give an example of a graph with at least4 vertices for which set of maximum weight edges SG is equal to the maximum weight spanning tree TG of graph G.

b.

Program Plan Intro

To give an example of a graph with at least4 vertices for which set of maximum weight edges SG is not equal to the maximum weight spanning tree TG of graph G.

c.

Program Plan Intro

To prove that SG⊆TG for any graph G.

d.

Program Plan Intro

To prove that w(SG)⊆w(TG)/2 for any graph G.

e.

Program Plan Intro

To give an O(V+E) -time algorithm to compute a 2-approximation to the maximum spanning tree.

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Answer 3

Question

Chapter 35, Problem 6P

a.

Program Plan Intro

To give an example of a graph with at least4 vertices for which set of maximum weight edges SG is equal to the maximum weight spanning tree TG of graph G.

b.

Program Plan Intro

To give an example of a graph with at least4 vertices for which set of maximum weight edges SG is not equal to the maximum weight spanning tree TG of graph G.

c.

Program Plan Intro

To prove that SG⊆TG for any graph G.

d.

Program Plan Intro

To prove that w(SG)⊆w(TG)/2 for any graph G.

e.

Program Plan Intro

To give an O(V+E) -time algorithm to compute a 2-approximation to the maximum spanning tree.

Expert Solution & Answer

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Answer 4

Question

Chapter 35, Problem 6P

a.

Program Plan Intro

To give an example of a graph with at least4 vertices for which set of maximum weight edges SG is equal to the maximum weight spanning tree TG of graph G.

b.

Program Plan Intro

To give an example of a graph with at least4 vertices for which set of maximum weight edges SG is not equal to the maximum weight spanning tree TG of graph G.

c.

Program Plan Intro

To prove that SG⊆TG for any graph G.

d.

Program Plan Intro

To prove that w(SG)⊆w(TG)/2 for any graph G.

e.

Program Plan Intro

To give an O(V+E) -time algorithm to compute a 2-approximation to the maximum spanning tree.

Expert Solution & Answer

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Answer 5

Chapter 35, Problem 6P

a.

Program Plan Intro

To give an example of a graph with at least4 vertices for which set of maximum weight edges SG is equal to the maximum weight spanning tree TG of graph G.

b.

Program Plan Intro

To give an example of a graph with at least4 vertices for which set of maximum weight edges SG is not equal to the maximum weight spanning tree TG of graph G.

c.

Program Plan Intro

To prove that SG⊆TG for any graph G.

d.

Program Plan Intro

To prove that w(SG)⊆w(TG)/2 for any graph G.

e.

Program Plan Intro

To give an O(V+E) -time algorithm to compute a 2-approximation to the maximum spanning tree.

Answer 6

Chapter 35, Problem 6P

a.

Program Plan Intro

To give an example of a graph with at least4 vertices for which set of maximum weight edges SG is equal to the maximum weight spanning tree TG of graph G.

Answer 7

Chapter 35, Problem 6P

a.

Program Plan Intro

To give an example of a graph with at least4 vertices for which set of maximum weight edges SG is equal to the maximum weight spanning tree TG of graph G.

Answer 8

b.

Program Plan Intro

To give an example of a graph with at least4 vertices for which set of maximum weight edges SG is not equal to the maximum weight spanning tree TG of graph G.

Answer 9

b.

Program Plan Intro

To give an example of a graph with at least4 vertices for which set of maximum weight edges SG is not equal to the maximum weight spanning tree TG of graph G.

Answer 10

c.

Program Plan Intro

To prove that SG⊆TG for any graph G.

Answer 11

c.

Program Plan Intro

To prove that SG⊆TG for any graph G.

Answer 12

d.

Program Plan Intro

To prove that w(SG)⊆w(TG)/2 for any graph G.

Answer 13

d.

Program Plan Intro

To prove that w(SG)⊆w(TG)/2 for any graph G.

Answer 14

e.

Program Plan Intro

To give an O(V+E) -time algorithm to compute a 2-approximation to the maximum spanning tree.

Answer 15

e.

Program Plan Intro

To give an O(V+E) -time algorithm to compute a 2-approximation to the maximum spanning tree.

Answer 16

Expert Solution & Answer

Answer 17

Expert Solution & Answer

Answer 18

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Answer 19

Ch. 35.1 - Prob. 1E Ch. 35.1 - Prob. 2E Ch. 35.1 - Prob. 3E Ch. 35.1 - Prob. 4E Ch. 35.1 - Prob. 5E Ch. 35.2 - Prob. 1E Ch. 35.2 - Prob. 2E Ch. 35.2 - Prob. 3E Ch. 35.2 - Prob. 4E Ch. 35.2 - Prob. 5E

To give an example of a graph with at least4 vertices for which set of maximum weight edges S G is equal to the maximum weight spanning tree T G of graph G.

Solution Summary: The author explains the greedy method of obtaining the maximum spanning tree of the graph by selecting the edge with maximum weight from remaining edges repeatedly.

Concept explainers

To give an example of a graph with at least4 vertices for which set of maximum weight edges SG is equal to the maximum weight spanning tree TG of graph G.

To give an example of a graph with at least4 vertices for which set of maximum weight edges SG is not equal to the maximum weight spanning tree TG of graph G.

To prove that SG⊆TG for any graph G.

To prove that w(SG)⊆w(TG)/2 for any graph G.

To give an O(V+E) -time algorithm to compute a 2-approximation to the maximum spanning tree.

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Chapter 35 Solutions