Given a directed graph $G$ and one of its vertices $u$, is there a polynomial time algorithm answering the following question:
Is there a chordless cycle (induced cycle) of $G$ containing the vertex $u$?
There exists algorithms to enumerate all chordless cycles but they are not polynomial see (http://sma.uni.lu/bisdorff/ChordlessCircuits/documents/chordlessCircuits.pdf).
I could not find any web page or research article answering this question.
Note that the question consider a directed graph and all cycle have consistent orientation.
In undirected graphs, for any cycle $[u, v_0, ..., v_k]$ and a chord $\{v_i, v_j\}$, $[u, v_0, ..., v_i, ..., v_j, ..., v_k]$ is a "smaller" cycle containing, after repeating this process for all chords, the algorithm will end up with a chordless cycle of G contaning u.
This does not work anymore in directed graphs because of the orientation of the chords. I tried to solve it with a tunned BREADTH FIRST SEARCH. If it found a cycle, this cycle is chordless but it did not unsure that it would find a chordless cycle containing u if such a cycle existed. I tried to improve it so would always find it but then I think it beacame exponential.
I recently found an article answer my question :
Anna Lubiw: A note on odd/even cycles. Discrete Applied Mathematics 22(1): 87-92 (1988) https://doi.org/10.1016/0166-218X(88)90125-4
It appears that the problem is NP-complete and thus there is no polynomial time algorithm to answer my question unless P equals NP.