# Snakes and Ladders: a Treewidth Story

Let G be an undirected graph. We say that G contains a ladder of length k if the 2 × (k+1) grid graph is an induced subgraph of G that is only connected to the rest of G via its four cornerpoints. We prove that if all the ladders contained in G are reduced to length 4, the treewidth remains unchanged (and that this bound is tight). Our result indicates that, when computing the treewidth of a graph, long ladders can simply be reduced, and that minimal forbidden minors for bounded treewidth graphs cannot contain long ladders. Our result also settles an open problem from algorithmic phylogenetics: the common chain reduction rule, used to simplify the comparison of two evolutionary trees, is treewidth-preserving in the display graph of the two trees.

• 25 publications
• 17 publications
• 5 publications
• 5 publications
• 7 publications
research
09/04/2018

### Treewidth of display graphs: bounds, brambles and applications

Phylogenetic trees and networks are leaf-labelled graphs used to model e...
research
03/11/2020

### Induced subgraphs of bounded treewidth and the container method

A hole in a graph is an induced cycle of length at least 4. A hole is lo...
research
12/21/2020

### Recoloring graphs of treewidth 2

Two (proper) colorings of a graph are adjacent if they differ on exactly...
research
04/30/2019

### Improved bounds for the excluded-minor approximation of treedepth

Treedepth, a more restrictive graph width parameter than treewidth and p...
research
03/23/2023

### Clustered independence and bounded treewidth

A set S⊆ V of vertices of a graph G is a c-clustered set if it induces a...
research
08/05/2020

### Constant Congestion Brambles

A bramble in an undirected graph G is a family of connected subgraphs of...
research
11/09/2018

### Minimizing and Computing the Inverse Geodesic Length on Trees

The inverse geodesic length (IGL) of a graph G=(V,E) is the sum of inver...