  For the Odd Cycle Transversal problem, the task is to find a small set $S$ of
vertices in a graph that intersects every cycle of odd length. The Subset Odd
Cycle Transversal problem requires S to intersect only those odd cycles that
include a vertex of a distinguished vertex subset $T$. If we are given weights
for the vertices, we ask instead that $S$ has small weight: this is the problem
Weighted Subset Odd Cycle Transversal. We prove an almost-complete complexity
dichotomy for Weighted Subset Odd Cycle Transversal for graphs that do not
contain a graph $H$ as an induced subgraph. Our general approach can also be
used for Weighted Subset Feedback Vertex Set, which enables us to generalize a
recent result of Papadopoulos and Tzimas.
