Forbidden Induced Subgraphs and the Łoś-Tarski Theorem

08/02/2020

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by   Yijia Chen, et al.

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Let 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 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.