Forbidden Induced Subgraphs and the Łoś-Tarski Theorem

Let C be a class of finite and infinite graphs that is closed under induced subgraphs. The wellknown Łoś-Tarski Theorem from classical model theory implies that C is definable in first-order logic (FO) by a sentence ϕ if and only if 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 ϕ the corresponding forbidden induced subgraphs. We show that this machinery fails on finite graphs.

-There is a class 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 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 ϕ, which defines C , and the size of the characterization cannot be bounded by f (|ϕ|) for any computable function f .

Besides their importance in graph theory, the above results also significantly strengthen similar known results for arbitrary structures.