Excluding a Forest Induced Minor

TL;DR

This paper establishes an induced minor analogue of the Forest Minor theorem, characterizing when certain $H$-induced-minor-free graphs have bounded pathwidth, advancing the understanding of graph structure and sparsity.

Contribution

It introduces a characterization of $H$-induced-minor-free graphs with bounded pathwidth, extending the Forest Minor theorem to induced substructures and classifying forests in this context.

Findings

Characterization of $H$-induced-minor-free graphs with bounded pathwidth.

Identification of a class $ extit{F}$ of forests related to induced minors.

Progress toward classifying graphs $H$ for bounded treewidth in sparse classes.

Abstract

In the first paper of the Graph Minors series [JCTB '83], Robertson and Seymour proved the Forest Minor theorem: the $H$-minor-free graphs have bounded pathwidth if and only if $H$ is a forest. In recent years, considerable effort has been devoted to understanding the unavoidable induced substructures of graphs with large pathwidth or large treewidth. In this paper, we give an induced counterpart of the Forest Minor theorem: for any $t \geqslant 2$, the $K_{t,t}$-subgraph-free $H$-induced-minor-free graphs have bounded pathwidth if and only if $H$ belongs to a class $\mathcal F$ of forests, which we describe as the induced minors of two (very similar) infinite parameterized families. This constitutes a significant step toward classifying the graphs $H$ for which every weakly sparse $H$-induced-minor-free class has bounded treewidth. Our work builds on the theory of constellations developed in the Induced Subgraphs and Tree Decompositions series.