Kruskal’s Minimum Spanning Tree Algorithm & Union-Find ...
in any MST of G. Suppose the theorem is false. Let T be a MST that contains e. Deleting e from T partitions vertices into 2 sets: S (that contains u) and V S (that contains v). Cycle C must have some other edge f that goes from S and V S. Replacing e by f produces a lower cost tree, contradicting that T is an MST.
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