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Prove that an edge-weighted graph GG has a unique minimum spanning tree if and only if the following conditions hold For any partition of the

Prove that an edge-weighted graph GG has a unique minimum spanning tree if and only if the following conditions hold

For any partition of the vertices of GG into two subsets, the minimum-weight edge with one endpoint in each subset is unique.

The maximum-weight edge in any cycle of GG is unique.

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