Given a directed graph $G$ and a vertex $v$, how can we enumerate all simple cycles that pass through $v$?
I found a question that describes how to enumerate all simple cycles in $G$, but I want only the cycles that pass through $v$, so this question is a bit different. The total number of cycles in $G$ can be exponentially larger than the number of cycles passing through $v$, so just enumerating all cycles in $G$ and keeping only those that go through $v$ might be inefficient.
I'm not asking to count the number of cycles; I want an algorithm to list them all. I would like an output-sensitive algorithm, where the running time is some low-order polynomial of the number of cycles that go through $v$. I know there can be exponentially many cycles, so there is no hope for an algorithm whose running time is polynomial in the size of the graph.
Motivation: This is a cleaned-up version of Given a directed graph and a vertex v, find all cycles that go through v?. I wanted to ask the more general version of the question.