Oriented expressions of graph properties

Abstract: 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.