Non-branching tree-decompositions

TL;DR

This paper proves that graphs with bounded treewidth can be decomposed into more structured forms, facilitating the analysis of their minors, especially in 2-connected graphs with large path-width.

Contribution

It introduces a method to refine tree-decompositions to have desirable properties, enabling new results on minors in graphs with large path-width.

Findings

Graphs with bounded treewidth have refined tree-decompositions with special properties.

Large 2-connected graphs with high path-width contain specific minors.

The results set the stage for characterizing minors in complex graphs.

Abstract

We prove that if a graph has a tree-decomposition of width at most w, then it has a tree-decomposition of width at most w with certain desirable properties. We will use this result in a subsequent paper to show that every 2-connected graph of large path-width has a minor isomorphic to either a large tree with a vertex attached to every vertex of the tree or a large outerplanar graph.