Let C and D be digraphs. A mapping f: V (D)→ V (C) is a C-colouring if for every arc uv of D,
either f (u) f (v) is an arc of C or f (u)= f (v), and the preimage of every vertex of C induces an
acyclic subdigraph in D. We say that D is C-colourable if it admits a C-colouring and that D is
uniquely C-colourable if it is surjectively C-colourable and any two C-colourings of D differ
by an automorphism of C. We prove that if a digraph D is not C-colourable, then there exist
digraphs of arbitrarily large girth that are D-colourable but not C-colourable. Moreover, for …

We prove that for every digraph $ D $ and every choice of positive integers $ k $, $\ell $
there exists a digraph $ D^* $ with girth at least $\ell $ together with a surjective acyclic
homomorphism $\psi\colon D^*\to D $ such that:(i) for every digraph $ C $ of order at most $
k $, there exists an acyclic homomorphism $ D^*\to C $ if and only if there exists an acyclic
homomorphism $ D\to C $; and (ii) for every $ D $-pointed digraph $ C $ of order at most $ k
$ and every acyclic homomorphism $\varphi\colon D^*\to C $ there exists a unique acyclic …