Logical complexity of induced subgraph isomorphism for certain graph families
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 $\ell-1$.