Linked tree-decompositions into finite parts

TL;DR

This paper proves that graphs with finite-part tree-decompositions can be refined into linked, tight, and componental forms, leading to stronger structural results and characterizations for graphs excluding half-grid minors.

Contribution

It introduces a method to refine any finite-part tree-decomposition into a linked, tight, and componental form, enabling new structural insights and characterizations.

Findings

Existence of linked, tight, componental tree-decompositions for such graphs

Strengthening of results on graphs without half-grid minors

Unified proofs of classical characterizations

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.