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Someone claims that Kruskal's algorithm for finding minimum spanning tree can return different spanning trees for the same input graph $G$. Do you agree with the claim? If so, why? If not, argue briefly why the claim is incorrect.
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Yes I think the claim is true as

If Edge weights are not distinct then we can have multiple Spanning trees

Suppose Consider an example

here we choose edges with weight 1 then with 2 then with 3

In case of 3 if we consider both edges whose weight is 3 the a cycle is formed so we consider only 1 edge by this we can have 2 Spanning trees with same weights

Spanning tree 1 :

Spanning Tree 2:

Let me know if I'm wrong

by Boss (13.9k points)
edited
Yes, I agree with the claim made and for its proof,

Taking an instance where we have more than one similar edge weights could be the graph (not neccessarily) on which applying kruskal's algo could give us more than one M.S.T's .

NOTE:- The claim holds false everytime we have distinct edge weights.

Let me know if I was wrong.
by (107 points)

It depends on whether edge weight is distinct or not.

Case 1: Edge weight is distinct.

MST is unique so Kruskal's algorithm (as its always works correctly to find MST) cannot give multiple MST.

Case 2: Edge weight not distinct

Multiple MST of same weight are possible. In this case Kruskal's algorithm may return different MST.

https://en.wikipedia.org/wiki/Kruskal%27s_algorithm#Proof_of_correctness

by Boss (12.5k points)