Two-factors in orientated graphs with forbidden transitions

The instance of the problem of finding 2-factors in an orientated graph with forbidden transitions consists of an orientated graph G and for each vertex v of G , a graph H v of allowed transitions at v . Vertices of the graph H v are the edges incident to v . An orientated 2-factor F of G is called legal if all the transitions are allowed, i.e. for every vertex v , the two edges of F adjacent to it form an edge in H v . Deciding whether a legal 2-factor exists in G is NP-complete in general. We investigate the case when the graphs of allowed transitions are taken from some fixed class C . We provide an exact characterization of classes C so that the problem is NP-complete. In particular, we prove a dichotomy for this problem, i.e. that for any class C it is either polynomial or NP-complete.