# A lower bound on the tree-width of graphs with irrelevant vertices

## TL;DR

This paper establishes a single-exponential lower bound on the function relating tree-width to the existence of irrelevant vertices in graphs, including planar graphs, for the disjoint paths problem.

## Contribution

It provides the first single-exponential lower bound on the function f(k) for the disjoint paths problem, improving understanding of graph structure related to tree-width.

## Key findings

- Single-exponential lower bound on f(k) for general graphs
- Bound applies to planar graphs as well
- Enhances understanding of irrelevant vertices in graph minors

## Abstract

For their famous algorithm for the disjoint paths problem, Robertson and Seymour proved that there is a function $f$ such that if the tree-width of a graph $G$ with $k$ pairs of terminals is at least $f(k)$, then $G$ contains a solution-irrelevant vertex (Graph Minors. XXII., JCTB 2012). We give a single-exponential lower bound on $f$. This bound even holds for planar graphs.