We have a digraph D=(V,A) and its arc set A is partitioned into classes. We can flip the classes, which means changing the direction of all the arcs in the class.

Is there any result on when can we make the digraph acyclic with this operation?

Even such special case would be interesting for me such as:

- the underlying graph is planar, and
- each classes form a path of same length in the underlying graph
- and each of these undirected paths (the classes) has alternating orientation.

Thanks for the answers.