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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
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