Given a directed unweighted graph how can we traverse though maximum number of nodes given we can choose any node as the source?

[torrtuga]

This kind of problem might become NP Hard if the graph is not acyclic.
Possible solution that comes in mind is Topological sorting and Hamiltonian Path solution but after in depth thought could not see a possible solution.
Similar question has been asked in stack exchange 2 years back but no correct answer.

What is the maximum number of nodes I can traverse in an undirected graph visiting each node exactly once?

Hope the codechef community will be of help.