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FORBIDDEN INDUCED SUBGRAPHS AND THE ŁOŚ–TARSKI THEOREM

Part of: Model theory General logic

Published online by Cambridge University Press:  04 January 2024

YIJIA CHEN  and
JÖRG FLUM
YIJIA CHEN*
Affiliation:
DEPARTMENT OF COMPUTER SCIENCE SHANGHAI JIAO TONG UNIVERSITY SHANGHAI, CHINA
JÖRG FLUM
Affiliation:
MATHEMATISCHES INSTITUT UNIVERSITÄT FREIBURG FREIBURG, GERMANY E-mail: joerg.flum@math.uni-freiburg.de
*
E-mail: yijia.chen@cs.sjtu.edu.cn
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Abstract

Let $\mathscr {C}$ 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 $\mathscr {C}$ is definable in first-order logic by a sentence $\varphi $ if and only if $\mathscr {C}$ has a finite set of forbidden induced finite subgraphs. This result 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. This machinery fails on finite graphs as shown by our results:

  • – There is a class $\mathscr {C}$ of finite graphs that is definable in first-order logic 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 that can be characterized by a finite set of forbidden induced subgraphs, such a characterization cannot be computed from a first-order sentence $\varphi $ that 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 theorems for arbitrary structures.

Keywords

preservation theorem for graphsfinite model theory

MSC classification

Primary: 03B70: Logic in computer science 03C13: Finite structures

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Type
Article
Information
The Journal of Symbolic Logic , Volume 89 , Issue 2 , June 2024 , pp. 516 - 548
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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