  In the \textsc{Subset Feedback Vertex Set (Subset-FVS)} problem the input is
a graph $G$, a subset \(T\) of vertices of \(G\) called the `terminal'
vertices, and an integer $k$. The task is to determine whether there exists a
subset of vertices of cardinality at most $k$ which together intersect all
cycles which pass through the terminals. \textsc{Subset-FVS} generalizes
several well studied problems including \textsc{Feedback Vertex Set} and
\textsc{Multiway Cut}. This problem is known to be \NP-Complete even in split
graphs. Cygan et al. proved that \textsc{Subset-FVS} is fixed parameter
tractable (\FPT) in general graphs when parameterized by $k$ [SIAM J. Discrete
Math (2013)]. In split graphs a simple observation reduces the problem to an
equivalent instance of the $3$-\textsc{Hitting Set} problem with same solution
size. This directly implies, for \textsc{Subset-FVS} \emph{restricted to split
graphs}, (i) an \FPT algorithm which solves the problem in $\OhStar(2.076^k)$
time \footnote{The \(\OhStar()\) notation hides polynomial factors.}% for
\textsc{Subset-FVS} in Chordal % Graphs [Wahlstr\"om, Ph.D. Thesis], and (ii) a
kernel of size $\mathcal{O}(k^3)$. We improve both these results for
\textsc{Subset-FVS} on split graphs; we derive (i) a kernel of size
$\mathcal{O}(k^2)$ which is the best possible unless $\NP \subseteq \coNP/{\sf
poly}$, and (ii) an algorithm which solves the problem in time
$\mathcal{O}^*(2^k)$. Our algorithm, in fact, solves \textsc{Subset-FVS} on the
more general class of \emph{chordal graphs}, also in $\mathcal{O}^*(2^k)$ time.
