I have a directed graph and I want to find a path that visits every node exactly one time. I want to do this with a good complexity. Is this possible? And if yes, how?
Visit all nodes exactly once in a directed graph
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You are searching for a Hamiltonian path, which is a simple open path that contains each node exactly once.
Finding a Hamiltonian path in a given graph is NP-complete. In fact, determining whether a given (directed or undirected) graph contains a Hamiltonian path is already NP-complete (proven via reduction from e.g. the vertex cover problem).
If you still want to code it, here is an implementation on github. If you want a fast solution, maybe a heuristic is sufficient (for instance inspired by DNA molecules, or a solution that works fast on a subset of graphs. For instance, if you have a DAG, you can do a topological sort and then check if successive vertices are connected. If so, the topological sort gives a Hamiltonian path.