Oriented expressions of graph properties

TL;DR

This paper investigates when graph properties can be characterized by orientations avoiding finitely many oriented subgraphs, revealing that some hereditary classes, including graphs with no prime-length holes, cannot be finitely characterized this way.

Contribution

It provides necessary conditions for such characterizations and demonstrates an uncountable family of classes, including those with no prime-length holes, that lack finite oriented graph characterizations.

Findings

Certain hereditary graph classes cannot be characterized by finitely many oriented structures.

Necessary conditions for a graph property to have a finite oriented structure characterization are established.

The class of graphs with no holes of prime length does not admit a finite oriented characterization.

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.