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Induced subgraphs and tree decompositions II. Toward walls and their line graphs in graphs of bounded degree.

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Date

2024-01

Authors

Abrishami, Tara
Chudnovsky, Maria
Dibek, Cemil
Hajebi, Sepehr
Rzqzewski, Pawel
Spirkl, Sophie
Vuskovic, Kristina

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Volume Title

Publisher

Elsevier

Abstract

This paper is motivated by the following question: what are the unavoidable induced subgraphs of graphs with large treewidth? Aboulker et al. made a conjecture which answers this question in graphs of bounded maximum degree, asserting that for all k and Δ, every graph with maximum degree at most Δ and sufficiently large treewidth contains either a subdivision of the (k × k)-wall or the line graph of a subdivision of the (k × k)-wall as an induced subgraph. We prove two theorems supporting this conjecture, as follows. 1. For t ≥ 2, a t-theta is a graph consisting of two nonadjacent vertices and three internally vertex-disjoint paths between them, each of length at least t. A t-pyramid is a graph consisting of a vertex v, a triangle B disjoint from v and three paths starting at v and vertex-disjoint otherwise, each joining v to a vertex of B, and each of length at least t. We prove that for all k, t and Δ, every graph with maximum degree at most Δ and sufficiently large treewidth contains either a t-theta, or a t-pyramid, or the line graph of a subdivision of the (k × k)-wall as an induced subgraph. This affirmatively answers a question of Pilipczuk et al. asking whether every graph of bounded maximum degree and sufficiently large treewidth contains either a theta or a triangle as an induced subgraph (where a theta means a t-theta for some t ≥ 2). 2. A subcubic subdivided caterpillar is a tree of maximum degree at most three whose all vertices of degree three lie on a path. We prove that for every Δ and subcubic subdivided caterpillar T, every graph with maximum degree at most Δ and sufficiently large treewidth contains either a subdivision of T or the line graph of a subdivision of T as an induced subgraph.

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The final publication is available at Elsevier via https://doi.org/10.1016/j.jctb.2023.10.005 © 2024. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/

Keywords

induced subgraph, tree decomposition, treewidth

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Citation

URI

https://doi.org/10.1016/j.jctb.2023.10.005
 http://hdl.handle.net/10012/20107

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Waterloo Research