For a fixed finite family of graphs $\mathcal{F}$, the $\mathcal{F}$-Minor-Free Deletion problem takes as input a graph G and integer ℓ and asks whether a size-ℓ vertex set X exists such that $G-X$ is $\mathcal{F}$-minor-free. $\{K_2\}$-Minor-Free Deletion and $\{K_3\}$-Minor-Free Deletion encode Vertex Cover and Feedback Vertex Set respectively. When parameterized by the feedback vertex number of G these two problems are known to admit a polynomial kernelization. We show $\{P_3\}$-Minor-Free Deletion parameterized by the feedback vertex number is $\mathsf{MK}[\mathsf{2}]$-hard. This rules out the existence of a polynomial kernel assuming $\mathsf{NP}⊈\mathsf{coNP}/\mathsf{poly}$. Our hardness result generalizes to any $\mathcal{F}$ containing only graphs with a connected component of at least 3 vertices, using as parameter the vertex-deletion distance to treewidth $\min \mathrm{tw}(\mathcal{F})$, where $\min \mathrm{tw}(\mathcal{F})$ denotes the minimum treewidth of the graphs in $\mathcal{F}$. For all other families $\mathcal{F}$ we present a polynomial Turing kernelization. Our results extend to $\mathcal{F}$-Subgraph-Free Deletion.

对于固定的有限图族 $\mathcal{F}$， 这 $\mathcal{F}$-Minor免费缺失问题作为输入的曲线图G ^和整数ℓ并询问一个大小-是否ℓ顶点集合X存在，使得$G-X$ 是 $\mathcal{F}$-次要的。 $\{{ķ}_{\mathrm{2个}}\}$-次要删除和$\{{ķ}_3\}$-次要-自由删除分别对“顶点覆盖”和“反馈顶点集”进行编码。当由G的反馈顶点数进行参数化时，这两个问题已知可以进行多项式核化。我们展示$\{P_3\}$由反馈顶点编号参数化的-次要-自由删除为$\mathsf{MK}[\mathsf{2个}]$-难的。这排除了多项式内核的存在$\mathsf{NP}⊈\mathsf{协同NP}/\mathsf{聚}$。我们的硬度结果可以推广到任何$\mathcal{F}$ 仅包含具有至少3个顶点的连通分量的图，并使用到树宽的顶点删除距离作为参数 $\mathrm{分}\mathrm{tw}（\mathcal{F}）$， 在哪里 $\mathrm{分}\mathrm{tw}（\mathcal{F}）$ 表示图中的最小树宽 $\mathcal{F}$。对于其他所有家庭$\mathcal{F}$我们提出了多项式图灵核化。我们的结果扩展到$\mathcal{F}$-Subgraph-Free删除。