A unified treatment of linked and lean tree-decompositions

TL;DR

This paper presents a unified framework for the existence of linked and lean tree-decompositions, generalizing previous results and applying to graphs and matroids, with implications for connectivity and decomposition width.

Contribution

It provides a general theorem on linked and lean tree-decompositions, unifying and extending known results, including for matroids.

Findings

Every graph has a linked tree-decomposition of minimal width.

Every matroid admits a lean tree-decomposition of width tw(M).

The results unify and extend previous theorems in the field.

Abstract

There are many results asserting the existence of tree-decompositions of minimal width which still represent local connectivity properties of the underlying graph, perhaps the best-known being Thomas' theorem that proves for every graph $G$ the existence of a linked tree-decompositon of width tw$(G)$. We prove a general theorem on the existence of linked and lean tree-decompositions, providing a unifying proof of many known results in the field, as well as implying some new results. In particular we prove that every matroid $M$ admits a lean tree-decomposition of width tw$(M)$, generalizing the result of Thomas.