Quickly excluding an apex-forest
<|reference_start|>Quickly excluding an apex-forest: 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.<|reference_end|>
@article{hodor2024quickly,
    title={Quickly excluding an apex-forest},
    author={Jk{e}drzej Hodor, Hoang La, Piotr Micek, Cl'ement Rambaud},
    journal={arXiv preprint arXiv:2404.17306},
    year={2024},
    archivePrefix={arXiv},
    eprint={2404.17306},
    primaryClass={math.CO cs.DM}
}