We show that every graph $G$ excluding an apex-forest $H$ as a minor has layered pathwidth at most $|V(H)|-2$, and that every graph $G$ excluding an apex-linear forest (such as a fan) $H$ as a minor has layered treedepth at most $|V(H)|-2$. We further show that both bounds are optimal. These results improve on recent results of Hodor, La, Micek, and Rambaud (2025): The first result improves the previous best-known bound by a multiplicative factor of $2$, while the second strengthens a previous quadratic bound. In addition, we reduce from quadratic to linear the bound on the $S$-focused treedepth $\mathrm{td}(G,S)$ for graphs $G$ with a prescribed set of vertices $S$ excluding models of paths in which every branch set intersects~$S$.

翻译：我们证明，每个不包含顶点森林$H$作为子式的图$G$具有层路径宽度至多$|V(H)|-2$，且每个不包含顶点线性森林（如扇形图）$H$作为子式的图$G$具有层树深度至多$|V(H)|-2$。我们进一步证明这两个界都是最优的。这些结果改进了Hodor、La、Micek和Rambaud（2025年）的最新成果：第一个结果将先前已知最佳界改进了2倍乘因子，而第二个结果强化了先前的二次界。此外，对于具有指定顶点集$S$且排除每个分支集与$S$相交的路径模型的图$G$，我们将$S$聚焦树深度$\mathrm{td}(G,S)$的界从二次降低至线性。