Logical complexity of induced subgraph isomorphism for certain graph families

TL;DR

This paper investigates the logical complexity of expressing induced subgraph isomorphism properties in first-order logic, establishing bounds on the quantifier depth needed for various graph families and sizes.

Contribution

It provides new bounds on the quantifier depth of first-order sentences defining induced subgraph isomorphism for specific graph classes and sizes.

Findings

For every $\, ext{ell}\, extgreater= 4$, there exists an $ ext{ell}$-vertex graph with a first-order sentence of quantifier depth at most $ ext{ell}-1$.

The property of containing an induced subgraph isomorphic to a disjoint union of complete multipartite graphs requires at least $ ext{ell}$ quantifiers.

For graphs with up to 5 vertices, the minimal quantifier depth is at least $ ext{ell}-1$.

Abstract

We prove that, for every $\ell\geq 4$, there exists an $\ell$-vertex graph and a first order sentence having a quantifier depth at most $\ell-1$ defining the property of having an induced subgraph isomorphic to the given one. We prove that a first order sentence defining the property of containing an induced subgraph on $\ell$ vertices isomorphic to a given disjoint union of isomorphic complete multipartite graphs has a quantifier depth at least $\ell$. Finally, we prove that, for every graph on $\ell\leq 5$ vertices a sentence defining the property of containing an induced subgraph isomorphic to the given one has a quantifier depth at least $\ell-1$.