I continue to learn the complexity myself, currently I am interested in the complexity of space. I have read several books and tried some exercises as a practice. I would like to have your idea on the following problem.
Show that the problem of the existence of a cycle in a directed graph is a $NL-complete$ problem. To show that the problem is $NL-hard$, start from problem $s; t-connectivity$ and as an intermediate step, create a acyclic graph $G^a$ which is $s’; t’- connected$ if and only if the original graph $G$ is $s; t- connected$.
The author has set as hint: use the length of the paths of a vertex $x$ at a vertex $y$.