Abrishami, T., Chudnovsky, M. orcid.org/0000-0002-8920-4944 and Vušković, K. (2022) Induced subgraphs and tree decompositions I. Even-hole-free graphs of bounded degree. Journal of Combinatorial Theory, Series B, 157. pp. 144-175. ISSN 0095-8956
Abstract
Treewidth is a parameter that emerged from the study of minor closed classes of graphs (i.e. classes closed under vertex and edge deletion, and edge contraction). It in some sense describes the global structure of a graph. Roughly, a graph has treewidth k if it can be decomposed by a sequence of noncrossing cutsets of size at most k into pieces of size at most k+1. The study of hereditary graph classes (i.e. those closed under vertex deletion only) reveals a different picture, where cutsets that are not necessarily bounded in size (such as star cutsets, 2-joins and their generalization) are required to decompose the graph into simpler pieces that are structured but not necessarily bounded in size. A number of such decomposition theorems are known for complex hereditary graph classes, including even-hole-free graphs, perfect graphs and others. These theorems do not describe the global structure in the sense that a tree decomposition does, since the cutsets guaranteed by them are far from being noncrossing. They are also of limited use in algorithmic applications. We show that in the case of even-hole-free graphs of bounded degree the cutsets described in the previous paragraph can be partitioned into a bounded number of well-behaved collections. This allows us to prove that even-hole-free graphs with bounded degree have bounded treewidth, resolving a conjecture of Aboulker et al. (2021) [1]. As a consequence, it follows that many algorithmic problems can be solved in polynomial time for this class, and that even-hole-freeness is testable in the bounded degree graph model of property testing. In fact we prove our results for a larger class of graphs, namely the class of C4-free odd-signable graphs with bounded degree.
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Item Type: | Article | ||||||
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Copyright, Publisher and Additional Information: | © 2022 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: | Treewidth; Even-hole-free graphs | ||||||
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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) | ||||||
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Depositing User: | Symplectic Publications | ||||||
Date Deposited: | 12 Apr 2024 16:24 | ||||||
Last Modified: | 12 Apr 2024 16:24 | ||||||
Status: | Published | ||||||
Publisher: | Elsevier | ||||||
Identification Number: | https://doi.org/10.1016/j.jctb.2022.05.009 |