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. It is known that the problem of deciding whether a given digraph can be made acyclic using the push operation is NP-complete for general digraphs, and polynomial time solvable for multipartite tournaments. Here, we continue the study of deciding whether a digraph is acyclically pushable, 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) and we characterize, in terms of two forbidden subdigraphs, the chordal digraphs which can be made acyclic using the push operation. An infinite family of chordal bipartite digraphs which are not acyclically pushable is described. 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.