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Prove that if G = (V, E) is a tree and e ∈ E, then (V,...

Prove that if G = (V, E) is a tree and e ∈ E, then (V, E − {e}) is a forest of two trees.

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Answer #1

Answer:

Prove that if G = (V, E) is a tree and e ∈ E, then (V, E − {e}) is a forest of two trees:

Ler G=(V, E) is a tree.

Take e ∈ E. Now if e=xy.

i.e,

x and y are the endpoints of e.

If (V, E − {e}) is not a forest then it is a tree. but an x and y are adjacent vertices,

But then is a cycle that starts and ends at x.

But as, G=(V, E) is a tree, it can't contain a cycle.

So, our assumption that (V, E-{e}) is a forest that is a union of two trees.

(As we delete one edge and it connects only two points, only two new trees will be created).

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