  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.

Data Structures and Algorithms;Computational Complexity;Discrete Mathematics;Combinatorics