Parity linkage and the Erd\H{o}s-P\'osa property of odd cycles through prescribed vertices in highly connected graphs

We show the following for every sufficiently connected graph $G$, any vertex subset $S$ of $G$, and given integer $k$: there are $k$ disjoint odd cycles in $G$ each containing a vertex of $S$ or there is set $X$ of at most $2k-2$ vertices such that $G-X$ does not contain any odd cycle that contains a vertex of $S$. We prove this via an extension of Kawarabayashi and Reed's result about parity-$k$-linked graphs (Combinatorica 29, 215-225). From this result it is easy to deduce several other well known results about the Erd\H{o}s-P\'osa property of odd cycles in highly connected graphs. This strengthens results due to Thomassen (Combinatorica 21, 321-333), and Rautenbach and Reed (Combinatorica 21, 267-278), respectively.
Comment: 12 pages, referees' comments incorporated