In the Feedback Vertex Set problem, we aim to find a small set $S$ of vertices in a graph intersecting every cycle. The Subset Feedback Vertex Set problem requires $S$ to intersect only those cycles that include a vertex of some specified set $T$. We also consider the Weighted Subset Feedback Vertex Set problem, where each vertex $u$ has weight $w(u)>0$ and we ask that $S$ has small weight. By combining known NP-hardness results with new polynomial-time results we prove full complexity dichotomies for Subset Feedback Vertex Set and Weighted Subset Feedback Vertex Set for $H$-free graphs, that is, graphs that do not contain a graph $H$ as an induced subgraph.

翻译：在回馈 Vertex Set 问题中, 我们的目标是在每个周期交叉的图表中找到一小套小套的脊椎美元。 子集反馈 Vertex Set 问题要求 $S 只交叉那些包含特定设定的顶点的周期。 我们还考虑 加权子集反馈 Vertex Set 问题, 每个顶点美元重量为 $w( u) > 0 美元, 我们要求$S 的重量小。 通过将已知 NP- 硬度结果与新的多元时间结果相结合, 我们证明子集反馈 Vertex 和加权子集反馈Vertex 设置的完全复杂的二分位模型, 即不包含以 $H 为诱导子图的图表 $H 。