Induced subgraphs and path decompositions

TL;DR

This paper investigates the unavoidable induced subgraphs in graphs with large pathwidth, providing results for sparse graphs and classes excluding fixed minors, and characterizing hereditary classes with bounded pathwidth.

Contribution

It introduces new results on induced subgraphs in graphs of large pathwidth, especially for sparse graphs and minor-excluding classes, and characterizes hereditary classes with bounded pathwidth.

Findings

Graphs with large pathwidth contain subdivisions of large binary trees or their line graphs as induced subgraphs.

In minor-excluding classes, similar induced subgraph structures are guaranteed.

A characterization of hereditary classes with bounded pathwidth is provided.

Abstract

A graph $H$ is an induced subgraph of a graph $G$ if a graph isomorphic to $H$ can be obtained from $G$ by deleting vertices. Recently, there has been significant interest in understanding the unavoidable induced subgraphs for graphs of large treewidth. Motivated by this work, we consider the analogous problem for pathwidth: what are the unavoidable induced subgraphs for graphs of large pathwidth? While resolving this question in the general setting looks challenging, we prove various results for sparse graphs. In particular, we show that every graph with bounded maximum degree and sufficiently large pathwidth contains a subdivision of a large complete binary tree or the line graph of a subdivision of a large complete binary tree as an induced subgraph. Similarly, we show that every graph excluding a fixed minor and with sufficiently large pathwidth contains a subdivision of a large complete binary tree or the line graph of a subdivision of a large complete binary tree as an induced subgraph. Finally, we present a characterisation for when a hereditary class defined by a finite set of forbidden induced subgraphs has bounded pathwidth.