# GATE | GATE CS 2019 | Question 47

Let G be any connection, 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 G, there is a unique minimum weight edge crossing the cut.

Which of the above two statements is/are TRUE?
(A) Neither I nor II
(B) I only
(C) II only
(D) Both I and II

Explanation: Statement (I) is always correct, as you need only (n-1) edges with n node graph for minimum spanning tree. You can prove that there will not be any other choice of edges other than first (n-1) lighter wight edges using Krushkal algorithm.

Note that converse of statement (I) is also true.

Statement (II) is also correct. Suppose MST is not unique, i.e., there exist T1 and T2 where both of them are MST and they are not identical. Suppose e1 ∈ T1 but e1 ∉ T2, if we remove e1 from T1, then we will have two trees with vertex sets V1 and V2. By problem 1 of HW #2, we know that e1 is a minimum cost edge in the cut between V1 and V2. Now consider T2, again problem 1 of HW #2, we know that T2 contains an edge e2 that is a minimum cost edge of the cut between V1 and V2. However, since e2 ≠ e1 we must have: c(e1) = c(e2) which is contradicting with the assumption that for every cut of the graph, the edge with the smallest cost across that cut is unique.

Note that converse of statement (II) may be true.

So, option (D) is correct.

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