The induced Erdős–Pósa conjecture for S-cycles

Let $G$ be a graph and $S\subseteq V(G)$. An $S$-cycle is a cycle containing a vertex of $S$, and an induced packing is a collection of cycles with no edge between distinct cycles. The induced Erdős–Pósa conjecture for $S$-cycles. There exists a function $f(k)=\mathcal{O}(k\log k)$ such that, for every positive integer $k$, every graph $G$, and every $S\subseteq V(G)$, $G$ contains either an induced packing of $k$ $S$-cycles or a set $X$ of at most $f(k)$ vertices such that $G-B_G(X,1)$ has no $S$-cycle. This extends the paper's induced cycle duality by requiring every packed or surviving cycle to meet a prescribed vertex set. No resolution is given in the supplied text.