Snakes and Ladders: a Treewidth Story

TL;DR

This paper proves that reducing long ladder structures in a graph to length 4 does not change its treewidth, simplifying computations and resolving an open problem in algorithmic phylogenetics.

Contribution

It establishes that long ladders can be safely reduced to length 4 without affecting treewidth, impacting graph minor theory and phylogenetics.

Findings

Long ladders can be reduced to length 4 without changing treewidth

Bound is tight; longer ladders do affect treewidth

The chain reduction rule preserves treewidth in phylogenetic display graphs

Abstract

Let $G$ be an undirected graph. We say that $G$ contains a ladder of length $k$ if the $2 \times (k+1)$ 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.