The strongly connected components are the same in a directed graph and its reverse

Lemma

Let be a directed graph and be its reverse. Both and have the same strongly connected components.

Proof

For any two vertices who are strongly connected in we need a path connecting to and path connecting to .

The reverse of the path connected to in and connects to in .

Therefore if two vertices are strongly connected in they are strongly connected in .

As this also gives us that if two vertices are strongly connected in they are strongly connected in .

Thus by the definition of strongly connected components they must be identical in and .