Forbidden Induced Subgraphs and the Łoś-Tarski Theorem

Abstract

Let $\mathcal{C}$ be a class of finite and infinite graphs that is closed under induced subgraphs. The well-known Łoś-Tarski Theorem from classical model theory implies that $\mathcal{C}$ is definable in first-order logic (FO) by a sentence $\phi$ if and only if $\mathcal{C}$ 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 $\phi$ the corresponding forbidden induced subgraphs. We show that this machinery fails on finite graphs. - There is a class $\mathcal{C}$ 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 $\mathcal{C}$ 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 $\phi$, which defines $\mathcal{C}$, and the size of the characterization cannot be bounded by $f(|\phi |)$ for any computable function $f$. Besides their importance in graph theory, the above results also significantly strengthen similar known results for arbitrary structures.