Explanation Formula used: In the Kruskal’s algorithm, find the edge with maximum weight and add to the tree. Also, repeated n − 1 times and the resulting tree will be a maximum spanning tree. Proof: Let G be a connected weighted graph. By the Kruskal’s algorithm on the graph produce a minimum spanning tree S . Assume that the edge selected by the algorithm are e 1 , e 2 , … e n − 1 in that order, the outcome of the Kruskal algorithm S that contains these edges. Let T be a minimum spanning tree of the graph G which conatin the edges e 1 , e 2 , … e k . If k = n − 1 , conclude that S = T and the algorithm has created the minimum spanning tree. If k is smaller than n − 1 , S is not a minimum spanning tree and the algorithm does not create minimum spanning tree. To prove this by contradiction. Find a minimum spanning tree which has become a larger value of k . Take the tree T and add an edge e k + 1 to it, the resulting graph must cover a circuit with e k + 1 . To find a edge e in the circuit that is not in S

Discrete Mathematics And Its Applications

8th Edition

ISBN: 9781260091991

Author: NA

Publisher: Mc graw hill

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Chapter 11.5, Problem 32E

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Kruskal’s algorithm produces minimum spanning trees.

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Chapter 11.5, Problem 31E

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Chapter 11 Solutions

Discrete Mathematics And Its Applications

Ch. 11.1 - Prob. 1E Ch. 11.1 - Vhich of these graphs are trees?Ch. 11.1 - Prob. 3E Ch. 11.1 - Prob. 4E Ch. 11.1 - Prob. 5E Ch. 11.1 - Prob. 6E Ch. 11.1 - Prob. 7E Ch. 11.1 - Prob. 8E Ch. 11.1 - Prob. 9E Ch. 11.1 - Prob. 10E

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Kruskal’s algorithm produces minimum spanning trees.

Solution Summary: The author explains how Kruskal's algorithm produces minimum spanning trees by finding the edge with maximum weight and adding to the tree.

To show: Kruskal’s algorithm produces minimum spanning trees.

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Chapter 11 Solutions

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Kruskal’s algorithm produces minimum spanning trees.