(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...

 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.

Extracted text: (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.