# On tree decompositions whose trees are minors

## TL;DR

This paper disproves a 2019 conjecture by showing that not all connected graphs, even with small treewidth, admit a tree decomposition with a tree that is a minor of the graph and has bounded width.

## Contribution

The paper provides a counterexample to the conjecture, demonstrating that such tree decompositions do not always exist for graphs of treewidth 2.

## Key findings

- Counterexample for graphs with treewidth 2
- Disproof of the conjecture for minor-based tree decompositions
- Shows limitations of tree decompositions with minor trees

## Abstract

In 2019, Dvo\v{r}\'{a}k asked whether every connected graph $G$ has a tree decomposition $(T, \mathcal{B})$ so that $T$ is a subgraph of $G$ and the width of $(T, \mathcal{B})$ is bounded by a function of the treewidth of $G$. We prove that this is false, even when $G$ has treewidth $2$ and $T$ is allowed to be a minor of $G$.