On the existence and non-existence of improper homomorphisms of oriented and $2$-edge-coloured graphs to reflexive targets

We consider non-trivial homomorphisms to reflexive oriented graphs in which some pair of adjacent vertices have the same image. Using a notion convexity for graphs, we study those that do not admit such homomorphisms. fully classify with tree-width $2$ and show it is NP-complete decide if graph admits an orientation does prove analogous results $2$-edge-coloured graphs. apply our on provide new tool chromatic number orientations planar -- long-standing open problem.