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Let G be any connected, weighted, undirected graph. I. G has a unique minimum spanning tree, if no two edges of G have the same weight. II. G has a unique minimum spanning tree, if, for every cut of G, there is a unique minimum-weight edge crossing the cut. Which of the above two statements is/are TRUE?

1. I only
2. II only
3. Both I and II
4. Neither I nor II

Correct answer: Both I and II

Solution

Both statements are true because the uniqueness of the minimum spanning tree is guaranteed by the absence of equal edge weights in the first case, and by the presence of a unique minimum-weight edge crossing every cut in the second case. These conditions ensure that there is only one way to construct the minimum spanning tree.

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