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### An Undirected Weighted Graph G **Figure Explanation:** The figure depicts an undirected weighted graph G with 6 vertices, labeled \( a, b, c, d, e, f \), and 9 edges with specified weights. The arrangement and weights are as follows: - \( a \) is positioned on the left. - \( f \) is above and to the right of \( a \). - \( b \) is below and to the right of \( f \), but above \( e \). - \( c \) is below and to the right of \( e \), but above \( d \). - The edges and their weights: - \( a \) and \( f \): 1 - \( f \) and \( e \): 4 - \( e \) and \( d \): 2 - \( f \) and \( a \): 3 - \( e \) and \( c \): 5 - \( c \) and \( a \): 7 - \( c \) and \( b \): 5 - \( a \) and \( b \): 6 **Tasks and Questions:** **(a) Use Prim’s algorithm to compute the minimum spanning tree for the weighted graph. Start the algorithm at vertex \( a \). Show the order in which the edges are added to the tree.** **(b) What is the minimum weight spanning tree for the weighted graph in the previous question subject to the condition that edge \((d, e)\) is in the spanning tree?** **(c) How would you generalize this idea? Suppose you are given a graph \( G \) and a particular edge \((u, v)\) in the graph. How would you alter Prim’s algorithm to find the minimum spanning tree subject to the condition that \((u, v)\) is in the tree?** --- For use in an educational context, the following details are necessary to understand and solve the tasks: **Figure 16:** This undirected weighted graph has the following layout and connections: - Vertices: \( a, b, c, d, e, f \) - Edges with weights: - \( a \) and \( f \): weight 1 - \( f \) and \( e \): weight 4 - \(

### An Undirected Weighted Graph G **Figure Explanation:** The figure depicts an undirected weighted graph G with 6 vertices, labeled \( a, b, c, d, e, f \), and 9 edges with specified weights. The arrangement and weights are as follows: - \( a \) is positioned on the left. - \( f \) is above and to the right of \( a \). - \( b \) is below and to the right of \( f \), but above \( e \). - \( c \) is below and to the right of \( e \), but above \( d \). - The edges and their weights: - \( a \) and \( f \): 1 - \( f \) and \( e \): 4 - \( e \) and \( d \): 2 - \( f \) and \( a \): 3 - \( e \) and \( c \): 5 - \( c \) and \( a \): 7 - \( c \) and \( b \): 5 - \( a \) and \( b \): 6 **Tasks and Questions:** **(a) Use Prim’s algorithm to compute the minimum spanning tree for the weighted graph. Start the algorithm at vertex \( a \). Show the order in which the edges are added to the tree.** **(b) What is the minimum weight spanning tree for the weighted graph in the previous question subject to the condition that edge \((d, e)\) is in the spanning tree?** **(c) How would you generalize this idea? Suppose you are given a graph \( G \) and a particular edge \((u, v)\) in the graph. How would you alter Prim’s algorithm to find the minimum spanning tree subject to the condition that \((u, v)\) is in the tree?** --- For use in an educational context, the following details are necessary to understand and solve the tasks: **Figure 16:** This undirected weighted graph has the following layout and connections: - Vertices: \( a, b, c, d, e, f \) - Edges with weights: - \( a \) and \( f \): weight 1 - \( f \) and \( e \): weight 4 - \(

Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
18th Edition
ISBN:9780079039897
Author:Carter
Publisher:Carter
Chapter1: Expressions And Functions
Section1.6: Relations
Problem 45PFA
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### An Undirected Weighted Graph G **Figure Explanation:** The figure depicts an undirected weighted graph G with 6 vertices, labeled \( a, b, c, d, e, f \), and 9 edges with specified weights. The arrangement and weights are as follows: - \( a \) is positioned on the left. - \( f \) is above and to the right of \( a \). - \( b \) is below and to the right of \( f \), but above \( e \). - \( c \) is below and to the right of \( e \), but above \( d \). - The edges and their weights: - \( a \) and \( f \): 1 - \( f \) and \( e \): 4 - \( e \) and \( d \): 2 - \( f \) and \( a \): 3 - \( e \) and \( c \): 5 - \( c \) and \( a \): 7 - \( c \) and \( b \): 5 - \( a \) and \( b \): 6 **Tasks and Questions:** **(a) Use Prim’s algorithm to compute the minimum spanning tree for the weighted graph. Start the algorithm at vertex \( a \). Show the order in which the edges are added to the tree.** **(b) What is the minimum weight spanning tree for the weighted graph in the previous question subject to the condition that edge \((d, e)\) is in the spanning tree?** **(c) How would you generalize this idea? Suppose you are given a graph \( G \) and a particular edge \((u, v)\) in the graph. How would you alter Prim’s algorithm to find the minimum spanning tree subject to the condition that \((u, v)\) is in the tree?** --- For use in an educational context, the following details are necessary to understand and solve the tasks: **Figure 16:** This undirected weighted graph has the following layout and connections: - Vertices: \( a, b, c, d, e, f \) - Edges with weights: - \( a \) and \( f \): weight 1 - \( f \) and \( e \): weight 4 - \(
Transcribed Image Text:### An Undirected Weighted Graph G **Figure Explanation:** The figure depicts an undirected weighted graph G with 6 vertices, labeled \( a, b, c, d, e, f \), and 9 edges with specified weights. The arrangement and weights are as follows: - \( a \) is positioned on the left. - \( f \) is above and to the right of \( a \). - \( b \) is below and to the right of \( f \), but above \( e \). - \( c \) is below and to the right of \( e \), but above \( d \). - The edges and their weights: - \( a \) and \( f \): 1 - \( f \) and \( e \): 4 - \( e \) and \( d \): 2 - \( f \) and \( a \): 3 - \( e \) and \( c \): 5 - \( c \) and \( a \): 7 - \( c \) and \( b \): 5 - \( a \) and \( b \): 6 **Tasks and Questions:** **(a) Use Prim’s algorithm to compute the minimum spanning tree for the weighted graph. Start the algorithm at vertex \( a \). Show the order in which the edges are added to the tree.** **(b) What is the minimum weight spanning tree for the weighted graph in the previous question subject to the condition that edge \((d, e)\) is in the spanning tree?** **(c) How would you generalize this idea? Suppose you are given a graph \( G \) and a particular edge \((u, v)\) in the graph. How would you alter Prim’s algorithm to find the minimum spanning tree subject to the condition that \((u, v)\) is in the tree?** --- For use in an educational context, the following details are necessary to understand and solve the tasks: **Figure 16:** This undirected weighted graph has the following layout and connections: - Vertices: \( a, b, c, d, e, f \) - Edges with weights: - \( a \) and \( f \): weight 1 - \( f \) and \( e \): weight 4 - \(
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