Excluding an apex-forest or a fan as quickly as possible

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~$S$.