  Several graph properties are characterized as the class of graphs that admit
an orientation avoiding finitely many oriented structures. For instance, if
$F_k$ is the set of homomorphic images of the directed path on $k+1$ vertices,
then a graph is $k$-colourable if and only if it admits an orientation with no
induced oriented graph in $F_k$. There is a fundamental question underlying
this kind of characterizations: given a graph property, $\mathcal{P}$, is there
a finite set of oriented graphs, $F$, such that a graph belongs to
$\mathcal{P}$ if and only if it admits an orientation with no induced oriented
graph in $F$? We address this question by exhibiting necessary conditions upon
certain graph classes to admit such a characterization. Consequently, we
exhibit an uncountable family of hereditary classes, for which no such finite
set exists. In particular, the class of graphs with no holes of prime length
belongs to this family.
