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.

翻译：我们给出了一个简短证明：对于任意至少包含两个顶点的顶冠森林 $X$，排除 $X$ 作为子式的图具有至多为 $2|V(X)|-3$ 的层路径宽度。这一结果改进了 Dujmović、Eppstein、Joret、Morin 和 Wood（SIDMA, 2020）的结论。我们的主要工具是一个关于排除森林作为根子式的图的结构性结果，该结果本身具有独立的研究价值。我们还为树深度和树宽开发了类似工具。本文讨论了这些结果对图中子式根模型的 Erdős–Pósa 性质的启示。