Do we always get the same set of edges after running Kruskal's algorithm on a single graph?

Question

I think it should be false because there may be more than one edge with the same weight.

Answer 1

Your are right. A simplest example would be a triangle with weight 1, 1, 1.

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You may find two definitive questions and answers at Are all MST minimum spanning trees reachable by Kruskal and Prim? and When is the minimum spanning tree for a graph not unique.