Quickly excluding an apex-forest

TL;DR

This paper provides a simplified proof that graphs excluding an apex-forest as a minor have bounded layered pathwidth, improving previous bounds and introducing new structural tools for related graph parameters.

Contribution

The authors present a concise proof establishing layered pathwidth bounds for apex-forest minor-free graphs and develop structural results for graphs excluding forests as rooted minors.

Findings

Graphs excluding apex-forests have layered pathwidth at most 2|V(X)|-3.

Structural results for graphs excluding forests as rooted minors.

Implications for Erdős-Pósa properties of rooted minors.

Abstract

We give a short proof that for every apex-forest $X$ on at least two vertices, graphs excluding $X$ as a minor have layered pathwidth at most $2|V(X)|-3$. This improves upon a result by Dujmovi\'c, Eppstein, Joret, Morin, and Wood (SIDMA, 2020). Our main tool is a structural result about graphs excluding a forest as a rooted minor, which is of independent interest. We develop similar tools for treedepth and treewidth. We discuss implications for Erd\H{o}s-P\'osa properties of rooted models of minors in graphs.