We consider a natural generalization of Vertex Cover: the Subset Vertex Cover problem, which is to decide for a graph $G=(V,E)$, a subset $T \subseteq V$ and integer $k$, if $V$ has a subset $S$ of size at most $k$, such that $S$ contains at least one end-vertex of every edge incident to a vertex of $T$. A graph is $H$-free if it does not contain $H$ as an induced subgraph. We solve two open problems from the literature by proving that Subset Vertex Cover is NP-complete on subcubic (claw,diamond)-free planar graphs and on $2$-unipolar graphs, a subclass of $2P_3$-free weakly chordal graphs. Our results show for the first time that Subset Vertex Cover is computationally harder than Vertex Cover (under P $\neq$ NP). We also prove new polynomial time results. We first give a dichotomy on graphs where $G[T]$ is $H$-free. Namely, we show that Subset Vertex Cover is polynomial-time solvable on graphs $G$, for which $G[T]$ is $H$-free, if $H = sP_1 + tP_2$ and NP-complete otherwise. Moreover, we prove that Subset Vertex Cover is polynomial-time solvable for $(sP_1 + P_2 + P_3)$-free graphs and bounded mim-width graphs. By combining our new results with known results we obtain a partial complexity classification for Subset Vertex Cover on $H$-free graphs.

翻译：我们考虑顶点覆盖问题的一个自然推广：子集顶点覆盖问题，即对图$G=(V,E)$、子集$T \subseteq V$和整数$k$，判断$V$是否存在一个大小不超过$k$的子集$S$，使得$S$包含至少一个与$T$中每个顶点相关联的边的端点。如果一个图不包含$H$作为其诱导子图，则称其为$H$-Free图。我们通过证明子集顶点覆盖问题在次立方（爪、钻石）-Free平面图和$2$-单极图（$2P_3$-Free弱弦图的一个子类）上是NP完全的，解决了文献中的两个开放问题。我们的结果首次表明（在P $\neq$ NP假设下）子集顶点覆盖问题在计算上比顶点覆盖问题更难。我们还证明了新的多项式时间结果。首先，我们给出了$G[T]$为$H$-Free的图的二分性结果：即当$H = sP_1 + tP_2$时，子集顶点覆盖问题在$G[T]$为$H$-Free的图$G$上可在多项式时间内求解，否则为NP完全。此外，我们证明了子集顶点覆盖问题在$(sP_1 + P_2 + P_3)$-Free图和有界mim-width图上可在多项式时间内求解。通过将我们的新结果与已知结果相结合，我们得到了$H$-Free图上子集顶点覆盖问题的部分复杂性分类。