# algorithms – Prove / Refute – topological topological transposition graph

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 $$v_i in the sort of $$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 $$G ^ T$$, $$v_j because now there is a path of $$v_j$$ at $$v_i$$, and if condition B is maintained, then the order $$v_j 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.