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Let's say that I have an MST, $T$. I pick an edge not in $T$ and change its weight, and add it to $T$ to create a cycle. Will removing the heaviest edge from that cycle result in an MST?

MST means minimum spanning tree of a graph. I came across these two posts:

and I follow both until the case where $w_{old}>w$ and $e\notin T$. They both say that deleting the heaviest edge will guarantee an MST, but I don't see how to prove that. The cycle property just says that IF you have an MST, it can't have an edge which is the heaviest edge in a cycle of the original graph $G$; it is NOT saying that IF you have a tree that doesn't contain an edge that happens to be the heaviest edge of some cycle in the original graph $G$, you are an MST.

To make the question more explicit in terms of the problem it was trying to solve, I will copy a part of the first link:

If its weight was reduced, add it to the original MST. This will create a cycle. Scan the cycle, looking for the heaviest edge (this could select the original edge again). Delete this edge.

I don't understand why this guarantees that we find an MST. Sure, we get a spanning tree but why does deleting this heaviest edge yield a MINIMUM spanning tree?

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  • $\begingroup$ Indeed, "deleting this heaviest edge yields a MINIMUM spanning tree" is not proved in either of the two posts you mentioned. A simple proof can be given if we can use the fact that the add-non-heavy-edge algorithm generates all MSTs and only MSTs. Here is the description of that algorithm. Let $G$ be a weighted graph. Start with an empty set. Iterate over all edges. Each edge is added to the set and, if a cycle is formed, remove one of its heaviest edges from the set. $\endgroup$
    – John L.
    Feb 11 '20 at 14:21
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The important part is that you start with an MST for the original graph. With this extra piece of information, you can build a proof by contradiction, as follows:

Construct the new tree as described (add the edge, check the cycle, remove the edge with the largest weight), now assume for contradiction that this tree is not an MST. This may be for several reasons:

  1. The real MST of the new graph differs in an edge that's not in the cycle. Then the original MST wasn't an MST to begin with, and we have a contradiction.
  2. The real MST differs in an edge that's in the cycle, but then, as the rest of the tree is the same, we can't have removed the heaviest edge from the cycle, and we have a contradiction again.

Therefore the new tree is an MST of the new graph.

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    $\begingroup$ "The real MST of the new graph". Does this proof rely on the assumption that there is only one MST? $\endgroup$
    – John L.
    Feb 11 '20 at 6:41
  • $\begingroup$ @JohnL.Good point, let me think about how to phrase the more complicated version readably... $\endgroup$ Feb 11 '20 at 9:29

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