 License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.CSL.2017.40
URN: urn:nbn:de:0030-drops-76841
URL: https://drops.dagstuhl.de/opus/volltexte/2017/7684/
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### On the First-Order Complexity of Induced Subgraph Isomorphism

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### Abstract

Given a graph F, let I(F) be the class of graphs containing F
as an induced subgraph. Let W[F] denote the minimum k such that
I(F) is definable in k-variable first-order logic. The recognition
problem of I(F), known as Induced Subgraph Isomorphism (for the pattern graph F), is solvable in time O(n^{W[F]}). Motivated by this fact, we are interested in determining or estimating the value of W[F]. Using Olariu's characterization of paw-free graphs, we show that I(K_3+e) is definable by a first-order sentence of quantifier depth 3, where K_3+e denotes the paw graph. This provides an example of a graph F with W[F] strictly less than the number of vertices in F.
On the other hand, we prove that W[F]=4 for all F on 4 vertices
except the paw graph and its complement. If F is a graph on t vertices, we prove a general lower bound W[F]>(1/2-o(1))t, where the function in the little-o notation approaches 0 as t increases. This bound holds true even for a related parameter W^*[F], which is
defined as the minimum k such that I(F) is definable in the k-variable
infinitary logic. We show that W^*[F] can be strictly less than W[F]. Specifically, W^*[P_4]=3 for P_4 being the path graph on 4 vertices.

### BibTeX - Entry

```@InProceedings{verbitsky_et_al:LIPIcs:2017:7684,
author =	{Oleg Verbitsky and Maksim Zhukovskii},
title =	{{On the First-Order Complexity of Induced Subgraph Isomorphism}},
booktitle =	{26th EACSL Annual Conference on Computer Science Logic (CSL 2017)},
pages =	{40:1--40:16},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-045-3},
ISSN =	{1868-8969},
year =	{2017},
volume =	{82},
editor =	{Valentin Goranko and Mads Dam},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
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