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arXiv · 1902.03648

Logical complexity of induced subgraph isomorphism for certain graph families

E. D. Kudryavtsev·M. V. Makarov·A. S. Shlychkova·M. E. Zhukovskii
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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 $\ell-1$.

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E. D. Kudryavtsev, M. V. Makarov, A. S. Shlychkova, M. E. Zhukovskii. 2019-02-10. Logical complexity of induced subgraph isomorphism for certain graph families. https://arxiv.org/abs/1902.03648

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