Given a directed graph $G$, a node $n \in G$, is it possible (besides bruteforcing for all possible solutions) to know if there exists a path starting from the node $n$ and such that we visit each node of the graph an odd number of times?
There are no other constraints except the starting point and the fact that I have to visit all nodes of the graph respecting the previous condition.
EDIT I'll make an example. Let's consider the graph in this picture: and assuming I want to start with the node 'a', there is no way to visit the graph passing through each node exactly 1 time (the only path is "a, c, a, b" passing through 'a' 2 times). If the starting node is 'c', instead, it's possible to do "c, a, b" and that satisfies the conditions.
Now obviously the graph isn't fully connected, but also changing the starting point can make things possible.
EDIT 2 I now realize the term "path" could be misleading. I just need to pass through each node an odd number of times.