Design a modification of the algorithm DFS to find a topological ordering of a given directed graph $($or to detect a directed cycle in the graph$)$ and prove of correctness of the solution.

Here's the DFS algorithm:

Algorithm DFS$\_$graph $($graph G$,$ vertex s$)$:

$(1)$ For each vertex u that belongs to V $($G$)$:

$(2)$ state$($u$)$ :$=$ not found

$(3)$ DFS$($s$)$

DFS $($vertex v$)$:

$(4)$ If state$($v$)=$ opened or closed:

$(5)$ return

$(6)$ state$($v$)$ :$=$ open

$(7)$ For each neighbor u of v:

$(8)$ DFS$($u$)$

$(9)$ state$($v$)$ :$=$ closed