(a) If T is a minimum spanning tree, then it is also a minimum spanning tree if you decrease the weight of some edge e ET by 1. (b) If T is a minimum spanning tree, then it is also a minimum spanning tree if some cut (A, B) and increase the weight of all edges crossing (A, B) by 1. you choose (c) If there is a unique minimum spanning tree, then for every cut (A, B) in G, there is a unique cheapest edge crossing (A, B). (d) If T is the unique minimum spanning tree, then T will also be the unique minimum spanning tree when you cube each edge weight. (e) If e is one of the cheapest edges crossing some cut (A, B) (maybe not the unique one), then e is in some minimum spanning tree.

(a) If T is a minimum spanning tree, then it is also a minimum spanning tree if you decrease the weight of some edge e ET by 1. (b) If T is a minimum spanning tree, then it is also a minimum spanning tree if some cut (A, B) and increase the weight of all edges crossing (A, B) by 1. you choose (c) If there is a unique minimum spanning tree, then for every cut (A, B) in G, there is a unique cheapest edge crossing (A, B). (d) If T is the unique minimum spanning tree, then T will also be the unique minimum spanning tree when you cube each edge weight. (e) If e is one of the cheapest edges crossing some cut (A, B) (maybe not the unique one), then e is in some minimum spanning tree.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

The following are statements about an undirected graph G = (V, E) with edge weights w(e). For each statement, either prove it, or disprove it with a briefly explained counterexample.

(a) If T is a minimum spanning tree, then it is also a minimum spanning tree if
the weight of some edge e ET by 1.
you
decrease
(b) If T is a minimum spanning tree, then it is also a minimum spanning tree if
some cut (A, B) and increase the weight of all edges crossing (A, B) by 1.
you
choose
(c) If there is a unique minimum spanning tree, then for every cut (A, B) in G, there is a
unique cheapest edge crossing (A, B).
(d) If T is the unique minimum spanning tree, then T will also be the unique minimum
spanning tree when you cube each edge weight.
(e) If e is one of the cheapest edges crossing some cut (A, B) (maybe not the unique one),
then e is in some minimum spanning tree.

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