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arXiv · 2405.06753

Linked tree-decompositions into finite parts

Sandra Albrechtsen·Raphael W. Jacobs·Paul Knappe·Max Pitz
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Abstract

We prove that every graph which admits a tree-decomposition into finite parts has a rooted tree-decomposition into finite parts that is linked, tight and componental. As an application, we obtain that every graph without half-grid minor has a lean tree-decomposition into finite parts, strengthening the corresponding result by Kriz and Thomas for graphs of finitely bounded tree-width. In particular, it follows that every graph without half-grid minor has a tree-decomposition which efficiently distinguishes all ends and critical vertex sets, strengthening results by Carmesin and by Elm and Kurkofka for this graph class. As a second application of our main result, it follows that every graph which admits a tree-decomposition into finite parts has a tree-decomposition into finite parts that displays all the ends of $G$ and their combined degrees, resolving a question of Halin from 1977. This latter tree-decomposition yields short, unified proofs of the characterisations due to Robertson, Seymour and Thomas of graphs without half-grid minor, and of graphs without binary tree subdivision.

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Sandra Albrechtsen, Raphael W. Jacobs, Paul Knappe, Max Pitz. 2024-05-10. Linked tree-decompositions into finite parts. https://arxiv.org/abs/2405.06753

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