  Given a digraph $D=(V,A)$ on $n$ vertices and a vertex $v\in V$, the
cycle-degree of $v$ is the minimum size of a set $S \subseteq V(D) \setminus
\{v\}$ intersecting every directed cycle of $D$ containing $v$. From this
definition of cycle-degree, we define the $c$-degeneracy (or cycle-degeneracy)
of $D$, which we denote by $\delta^*_c(D)$. It appears to be a nice
generalisation of the undirected degeneracy. In this work, using this new
definition of cycle-degeneracy, we extend several evidences for Cereceda's
conjecture to digraphs. The $k$-dicolouring graph of $D$, denoted by
$\mathcal{D}_k(D)$, is the undirected graph whose vertices are the
$k$-dicolourings of $D$ and in which two $k$-dicolourings are adjacent if they
differ on the colour of exactly one vertex.
  We show that $\mathcal{D}_k(D)$ has diameter at most
$O_{\delta^*_c(D)}(n^{\delta^*_c(D) + 1})$ (respectively $O(n^2)$ and
$(\delta^*_c(D)+1)n$) when $k$ is at least $\delta^*_c(D)+2$ (respectively
$\frac{3}{2}(\delta^*_c(D)+1)$ and $2(\delta^*_c(D)+1)$). This improves known
results on digraph redicolouring (Bousquet et al.).
  Next, we extend a result due to Feghali to digraphs, showing that
$\mathcal{D}_{d+1}(D)$ has diameter at most $O_{d,\epsilon}(n(\log n)^{d-1})$
when $D$ has maximum average cycle-degree at most $d-\epsilon$. We then show
that two proofs of Bonamy and Bousquet for undirected graphs can be extended to
digraphs. The first one uses the digrundy number of a digraph and the second
one uses the $\mathscr{D}$-width.
  Finally, we give a general theorem which makes a connection between the
recolourability of a digraph $D$ and the recolourability of its underlying
graph $UG(D)$. This result directly extends a number of results on planar graph
recolouring to planar digraph redicolouring.
