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Suppose we have a connected, undirected graph $G=(V,E)$, with a weight function $w\colon E\to R$ and let $T$ be a minimum spanning tree. Now assume that we remove a particular edge $e$. The exercise is to find a minimum spanning tree to the new graph.

I've thought of an algorithm: Look at the cut determined by $T$ and $e$, and then replace $e$ by the minimal weighted edge crossing that cut (if such an edge exists). But I don't know how to prove this works. Help?

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    I assume$e$is in T. So if I remove$e$from T, then two connected components are left, and they form a cut for the graph (i.e a partition of V into two subsets). That's what I meant in "the cut determined by$T$and e".2012-01-14

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