research-article

Graphs Identified by Logics with Counting

Authors:

Pascal Schweitzer,

Erkal SelmanAuthors Info & Claims

ACM Transactions on Computational Logic (TOCL), Volume 23, Issue 1

Article No.: 1, Pages 1 - 31

https://doi.org/10.1145/3417515

Published: 22 October 2021 Publication History

Abstract

We classify graphs and, more generally, finite relational structures that are identified by \(C^2\), that is, two-variable first-order logic with counting. Using this classification, we show that it can be decided in almost linear time whether a structure is identified by \(C^2\). Our classification implies that for every graph identified by this logic, all vertex-colored versions of it are also identified. A similar statement is true for finite relational structures.

We provide constructions that solve the inversion problem for finite relational structures in linear time. By a result due to Otto, this problem has been known to be polynomial-time solvable. For graphs, we conclude that every \(C^2\)-equivalence class contains a representative whose orbits are exactly the classes of the \(C^2\)-partition of its vertex set and which has a single automorphism witnessing this fact.

We show that such statements are not true for general \(k\) by providing examples of graphs of order linear in \(k\) which are identified by \(C^3\), but for which the orbit partition is strictly finer than the \(C^k\)-partition. We also construct identified graphs which have vertex-colored versions that are not identified by \(C^k\).

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Cited By

Guo JGavrilyuk APonomarenko I(2024)On the Weisfeiler–Leman Dimension of Permutation GraphsSIAM Journal on Discrete Mathematics10.1137/23M157501938:2(1915-1929)Online publication date: 21-Jun-2024
https://doi.org/10.1137/23M1575019
Grochow JLevet M(2023)On the Descriptive Complexity of Groups without Abelian Normal Subgroups (Extended Abstract)Electronic Proceedings in Theoretical Computer Science10.4204/EPTCS.390.12390(185-202)Online publication date: 30-Sep-2023
https://doi.org/10.4204/EPTCS.390.12

Index Terms

Graphs Identified by Logics with Counting
1. Mathematics of computing
  1. Discrete mathematics
    1. Graph theory
      1. Graph algorithms
2. Theory of computation
  1. Logic
    1. Finite Model Theory

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Published In

cover image ACM Transactions on Computational Logic

ACM Transactions on Computational Logic Volume 23, Issue 1

January 2022

237 pages

ISSN:1529-3785

EISSN:1557-945X

DOI:10.1145/3487995

Editor:
Anuj Dawar
University of Cambridge, United Kingdom

Issue’s Table of Contents

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected].

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 22 October 2021

Accepted: 01 August 2020

Revised: 01 March 2020

Received: 01 April 2019

Published in TOCL Volume 23, Issue 1

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Cited By

Guo JGavrilyuk APonomarenko I(2024)On the Weisfeiler–Leman Dimension of Permutation GraphsSIAM Journal on Discrete Mathematics10.1137/23M157501938:2(1915-1929)Online publication date: 21-Jun-2024
https://doi.org/10.1137/23M1575019
Grochow JLevet M(2023)On the Descriptive Complexity of Groups without Abelian Normal Subgroups (Extended Abstract)Electronic Proceedings in Theoretical Computer Science10.4204/EPTCS.390.12390(185-202)Online publication date: 30-Sep-2023
https://doi.org/10.4204/EPTCS.390.12

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