Forbidden Induced Subgraphs and the {\L}o\'s-Tarski Theorem

Let $\mathscr C$ be a class of finite and infinite graphs that is closed under induced subgraphs. The well-known {\L}o\'s-Tarski Theorem from classical model theory implies that $\mathscr C$ is definable in first-order logic (FO) by a sentence $\varphi$ if and only if $\mathscr C$ has a finite set of forbidden induced finite subgraphs. It provides a powerful tool to show nontrivial characterizations of graphs of small vertex cover, of bounded tree-depth, of bounded shrub-depth, etc. in terms of forbidden induced finite subgraphs. Furthermore, by the Completeness Theorem, we can compute from $\varphi$ the corresponding forbidden induced subgraphs. We show that this machinery fails on finite graphs. - There is a class $\mathscr C$ of finite graphs which is definable in FO and closed under induced subgraphs but has no finite set of forbidden induced subgraphs. - Even if we only consider classes $\mathscr C$ of finite graphs which can be characterized by a finite set of forbidden induced subgraphs, such a characterization cannot be computed from an FO-sentence $\varphi$, which defines $\mathscr C$, and the size of the characterization cannot be bounded by $f(|\varphi|)$ for any computable function $f$. Besides their importance in graph theory, the above results also significantly strengthen similar known results for arbitrary structures.