Pushing vertices in digraphs without long induced cycles

Given a digraph D and a subset X of vertices of D, pushing X in D means reversing the orientation of all arcs with exactly one end in X. We continue the study of deciding whether a digraph can be made acyclic using the push operation, focussing on special classes of well-structured digraphs. It is proved that the problem remains NP-complete even when restricted to the class of bipartite digraphs (i.e., oriented bipartite graphs). We characterize, in terms of two forbidden subdigraphs, the chordal digraphs which can be made acyclic using the push operation, and show that there is no similar characterization for the family of chordal bipartite digraphs. A polynomial algorithm, based on 2-SAT, for solving the problem for a subclass of the chordal bipartite digraphs is given. Finally, a characterization in terms of a finite number of forbidden subdigraphs, of the acyclically pushable bipartite permutation digraphs is given.