Complexity dichotomy for oriented homomorphism of planar graphs with large girth

We consider the complexity of oriented homomorphism and two of its variants, namely strong oriented homomorphism and pushable homomorphism, for planar graphs with large girth. In each case, we consider the smallest target graph such that the corresponding homomorphism is NP-complete. These target graphs T4T4, T5T5, and T6T6 have 4, 5, and 6 vertices, respectively. For i∈{4,5,6}i∈{4,5,6} and for every g, we prove that if there exists a (bipartite) planar oriented graph with girth at least g   that does not map to TiTi, then deciding homomorphism to TiTi is NP-complete for (bipartite) planar oriented graphs with girth at least g.