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Basically what the title says. Suppose an edge e is in every minimum spanning tree of G, does that means that e is a cut edge in G? Can I just find a counter example using the contraposition to solve this?

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As Gerry's answer points out, this is false (for weighted graphs). What is true is that if an edge is in every spanning tree (not necessarily minimum weight) of $G$, then it is a cut edge. – Srivatsan Nov 7 '11 at 4:44
@EdFox If you want to get someone's attention on this forum, you need to put @ in front of the person's username, as I've done for you here. – Austin Mohr Nov 7 '11 at 8:01
Could you write a small proof of that, @Srivatsan? Also how about if G is weighted again and e is the edge of maximum weight in G? I think the two questions have to be related. – Ed Fox Nov 7 '11 at 11:44
up vote 3 down vote accepted

If MST means minimal (weight) spanning tree (in a weighted graph), then the answer is, certainly not. Let $G$ be a complete graph wit one edge of weight 1 and lots of edges of weight 17. That weight 1 edge will be in every MST, but it's not a cut edge.

Unless I've misunderstood your definitions/conventions.

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