I must prove or refute the following statement:

*"Let $ G $ be a directed acyclic graph, and $ v_1v_2 … v_k $ a kind of topology $ G $.
then $ v_kv_ {k-1} … $ v_1 is a valid topological kind of the transposed graph $ G ^ T $. "*

(The algorithm used is the one presented in CLRS 22.4)

I think the statement is **correct**.

My logic is that for every two summits $ v_i, v_j $ at $ v_i <v_j $ in the sort of $ G $one of the following two conditions:

A) $ v_j $ is a deceased of $ v_i $ in $ G $, so in every kind of topology of $ G $, $ v_i $ will be before $ v_j $ in the genre.

B) or there is no way to $ v_i $ at $ v_j $ or the opposite, then $ v_i $ and $ v_j $ can change places in sorting, depending on the order of execution of the DFS $ G $.

From there, if condition A is fulfilled then in sorting $ G ^ T $, $ v_j <v_i $because now there is a path of $ v_j $ at $ v_i $, and if condition B is maintained, then the order $ v_j <v_i $ is valid for a kind of topology $ G ^ T $. And then the sorting $ v_kv_ {k-1} … $ v_1 is somehow valid $ G ^ T $.

Something is still wrong. I would appreciate any help or comment.