Abrishami, T., Chudnovsky, M., Dibek, C. et al. (4 more authors) (2024) Induced subgraphs and tree decompositions II. Toward walls and their line graphs in graphs of bounded degree. Journal of Combinatorial Theory: Series B, 164. pp. 371-403. ISSN 0095-8956
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 x k)-wall or the line graph of a subdivision of the (k x 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 x 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.
Metadata
Authors/Creators: |
|
||||
---|---|---|---|---|---|
Copyright, Publisher and Additional Information: | © 2023 Elsevier Inc. This is an author produced version of an article published in Journal of Combinatorial Theory, Series B. Uploaded in accordance with the publisher's self-archiving policy. | ||||
Keywords: | Induced Subgraph; Tree decomposition; Treewidth | ||||
Dates: |
|
||||
Institution: | The University of Leeds | ||||
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Computing (Leeds) | ||||
Funding Information: |
|
||||
Depositing User: | Symplectic Publications | ||||
Date Deposited: | 24 Oct 2023 08:56 | ||||
Last Modified: | 14 Nov 2023 12:36 | ||||
Status: | Published | ||||
Publisher: | Elsevier | ||||
Identification Number: | https://doi.org/10.1016/j.jctb.2023.10.005 |