Abrishami, T., Alecu, B., Chudnovsky, M. et al. (3 more authors) (Cover date: April 2024) Induced subgraphs and tree decompositions V. one neighbor in a hole. Journal of Graph Theory, 105 (4). pp. 542-561. ISSN 0364-9024
Abstract
What are the unavoidable induced subgraphs of graphs with large treewidth? It is well-known that the answer must include a complete graph, a complete bipartite graph, all subdivisions of a wall and line graphs of all subdivisions of a wall (we refer to these graphs as the “basic treewidth obstructions”). So it is natural to ask whether graphs excluding the basic treewidth obstructions as induced subgraphs have bounded treewidth. Sintiari and Trotignon answered this question in the negative. Their counterexamples, the so-called “layered wheels,” contain wheels, where a wheel consists of a hole (i.e., an induced cycle of length at least four) along with a vertex with at least three neighbors in the hole. This leads one to ask whether graphs excluding wheels and the basic treewidth obstructions as induced subgraphs have bounded treewidth. This also turns out to be false due to Davies' recent example of graphs with large treewidth, no wheels and no basic treewidth obstructions as induced subgraphs. However, in Davies' example there exist holes and vertices (outside of the hole) with two neighbors in them. Here we prove that a hole with a vertex with at least two neighbors in it is inevitable in graphs with large treewidth and no basic obstruction. Our main result is that graphs in which every vertex has at most one neighbor in every hole (that does not contain it) and with the basic treewidth obstructions excluded as induced subgraphs have bounded treewidth.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2023 The Authors. Journal of Graph Theory published by Wiley Periodicals LLC. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. |
Keywords: | induced subgraph; tree decomposition; treewidth |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Computing (Leeds) |
Funding Information: | Funder Grant number EPSRC (Engineering and Physical Sciences Research Council) EP/V002813/1 |
Depositing User: | Symplectic Publications |
Date Deposited: | 26 Sep 2023 09:43 |
Last Modified: | 21 Mar 2024 16:34 |
Status: | Published |
Publisher: | Wiley |
Identification Number: | 10.1002/jgt.23055 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:203673 |