Greedy-Grow Lemma

Lemma: If T is a subset of some MST, and C is some cut that doesn't share any edges with T, then there is some MST containing T and the minimum cost edge in C.

Proof: Very standard trick. Let be the MST that contains T. If , we're done. If not, consider the set of edges . It must have a cycle including and some other edge e (maybe more than one) in C. By definition, . Delete e from .

• Is the resulting set still a spanning tree?
• What is its cost compared to ?