For a fixed finite collection of graphs $\mathcal{F}$, the $\mathcal{F}$-Minor-Free Deletion problem asks, given a graph $G$ and an integer $k$, whether there is a set $S\subseteq V(G)$ with $|S|\leq k$ such that $G-S$ contains no graph of $\mathcal{F}$ as a minor. This generalizes Vertex Cover, Feedback Vertex Set, and Vertex Planarization, among many others. Baste, Sau, and Thilikos [JCSS 2020, SICOMP 2023] showed that, assuming the Exponential Time Hypothesis (ETH), there are collections $\mathcal{F}$ for which the problem cannot be solved in time $2^{o(\mathrm{tw}\log\mathrm{tw})}n^{O(1)}$ on $n$-vertex graphs of treewidth $\mathrm{tw}$, and gave a matching algorithm. We ask whether this super-exponential dependence persists under more restrictive structural parameters. First, we strengthen their lower bound. Assuming ETH, for every fixed $h\geq 6$, $\{P_h\}$-Minor-Free Deletion, where $P_h$ is the path on $h$ vertices, cannot be solved in time $2^{o(t\log t)}n^{O(1)}$, where $t=\mathrm{fvs}+\mathrm{td}$ is the sum of the feedback vertex set number and treedepth of the input graph. These two parameters are incomparable, but both individually upper bound the treewidth. Together with the $2^{O(\mathrm{tw}\log\mathrm{tw})}n^{O(1)}$-time algorithm of Baste, Sau, and Thilikos [SICOMP 2023], our lower bound is tight. In contrast, we show that asymptotically faster algorithms are possible when parameterized by the vertex cover number $\mathrm{vc}$. Specifically, for any finite family $\mathcal{F}$, $\mathcal{F}$-Minor-Free Deletion can be solved in time $2^{O(\mathrm{vc})}n^{O(1)}$. Taken together, our results show that the boundary between single-exponential and super-exponential complexity for this problem lies between the vertex cover number and the feedback vertex set number (plus treedepth).

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