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In the \textsc{Subset Feedback Vertex Set (Subset-FVS)} problem the input isa 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 asubset of vertices of cardinality at most $k$ which together intersect allcycles which pass through the terminals. \textsc{Subset-FVS} generalizesseveral well studied problems including \textsc{Feedback Vertex Set} and\textsc{Multiway Cut}. This problem is known to be \NP-Complete even in splitgraphs. Cygan et al. proved that \textsc{Subset-FVS} is fixed parametertractable (\FPT) in general graphs when parameterized by $k$ [SIAM J. DiscreteMath (2013)]. In split graphs a simple observation reduces the problem to anequivalent instance of the $3$-\textsc{Hitting Set} problem with same solutionsize. This directly implies, for \textsc{Subset-FVS} \emph{restricted to splitgraphs}, (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) akernel 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 $\NP \subseteq \coNP/{\sfpoly}$, and (ii) an algorithm which solves the problem in time$\mathcal{O}^*(2^k)$. Our algorithm, in fact, solves \textsc{Subset-FVS} on themore general class of \emph{chordal graphs}, also in $\mathcal{O}^*(2^k)$ time.

Subset Feedback Vertex Set in Chordal and Split Graphs