  It has been shown by Bokal et al. that deciding 2-colourability of digraphs
is an NP-complete problem. This result was later on extended by Feder et al. to
prove that deciding whether a digraph has a circular $p$-colouring is
NP-complete for all rational $p>1$. In this paper, we consider the complexity
of corresponding decision problems for related notions of fractional colourings
for digraphs and graphs, including the star dichromatic number, the fractional
dichromatic number and the circular vertex arboricity. We prove the following
results:
  Deciding if the star dichromatic number of a digraph is at most $p$ is
NP-complete for every rational $p>1$.
  Deciding if the fractional dichromatic number of a digraph is at most $p$ is
NP-complete for every $p>1, p \neq 2$.
  Deciding if the circular vertex arboricity of a graph is at most $p$ is
NP-complete for every rational $p>1$.
  To show these results, different techniques are required in each case. In
order to prove the first result, we relate the star dichromatic number to a new
notion of homomorphisms between digraphs, called circular homomorphisms, which
might be of independent interest. We provide a classification of the
computational complexities of the corresponding homomorphism colouring problems
similar to the one derived by Feder et al. for acyclic homomorphisms.
