On the complexity of digraph packings

Let G be a fixed collection of digraphs. Given a digraph H, a G-packing of H is a collection of vertex disjoint subgraphs of H, each isomorphic to a member of G. For undirected graphs, Loebl and Poljak have completely characterized the complexity of deciding the existence of a perfect G-packing, in the case that G consists of two graphs one of which is a single edge on two vertices. We characterize G-packing where G consists of two digraphs one of which is a single arc on two vertices.