Quickly excluding an apex-forest
J\k{e}drzej Hodor, Hoang La, Piotr Micek, Cl\'ement Rambaud

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 on at least two vertices, graphs excluding as a minor have layered pathwidth at most . 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.
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Taxonomy
TopicsAdvanced Graph Theory Research · Complexity and Algorithms in Graphs · Parallel Computing and Optimization Techniques
