research-article

Planar Graphs Have Bounded Queue-Number

Authors:
Vida Dujmović

University of Ottawa, Ottawa, Canada

University of Ottawa, Ottawa, Canada
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,
Gwenaël Joret

Université Libre de Bruxelles, Brussels, Belgium

Université Libre de Bruxelles, Brussels, Belgium
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,
Piotr Micek

Jagiellonian University, Kraków, Poland

Jagiellonian University, Kraków, Poland
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,
Pat Morin

Carleton University, Ottawa, Canada

Carleton University, Ottawa, Canada
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,
Torsten Ueckerdt

Karlsruhe Institute of Technology, Karlsruhe, Germany

Karlsruhe Institute of Technology, Karlsruhe, Germany
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,
David R. Wood

Monash University, Melbourne, Australia

Monash University, Melbourne, Australia
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Authors Info & Claims

Journal of the ACM Volume 67 Issue 4Article No.: 22pp 1–38https://doi.org/10.1145/3385731

Published:06 August 2020Publication History

Journal of the ACM

Abstract

We show that planar graphs have bounded queue-number, thus proving a conjecture of Heath et al. [66] from 1992. The key to the proof is a new structural tool called layered partitions, and the result that every planar graph has a vertex-partition and a layering, such that each part has a bounded number of vertices in each layer, and the quotient graph has bounded treewidth. This result generalises for graphs of bounded Euler genus. Moreover, we prove that every graph in a minor-closed class has such a layered partition if and only if the class excludes some apex graph. Building on this work and using the graph minor structure theorem, we prove that every proper minor-closed class of graphs has bounded queue-number.

Layered partitions have strong connections to other topics, including the following two examples. First, they can be interpreted in terms of strong products. We show that every planar graph is a subgraph of the strong product of a path with some graph of bounded treewidth. Similar statements hold for all proper minor-closed classes. Second, we give a simple proof of the result by DeVos et al. [31] that graphs in a proper minor-closed class have low treewidth colourings.

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Index Terms

Planar Graphs Have Bounded Queue-Number
1. Mathematics of computing

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Published in
Journal of the ACM Volume 67, Issue 4
August 2020
265 pages
ISSN:0004-5411
EISSN:1557-735X
DOI:10.1145/3403612
Editor:
Éva Tardos
Cornell University
Issue’s Table of Contents
Copyright © 2020 ACM
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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Publication History
- Published: 6 August 2020
- Online AM: 7 May 2020
- Accepted: 1 February 2020
- Received: 1 May 2019
Published in jacm Volume 67, Issue 4

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Graph theory
graph minor
graph product
planar graph
queue layout
queue-number
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Planar Graphs Have Bounded Queue-Number

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On the Maximum Number of Cliques in a Graph

Layout of Graphs with Bounded Tree-Width

Equitable colorings of planar graphs without short cycles