In the 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. SUBSET-FVS generalizes several well studied problems including FEEDBACK VERTEX SET and MULTIWAY CUT. This problem is known to be NP-Complete even in split graphs. Cygan et al. proved that 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-HITTING SET problem with same solution size. This directly implies, for SUBSET-FVS restricted to split graphs, (i) an FPTalgorithm which solves the problem in O (2.076 k ) time 6 [Wahlström, Ph.D. Thesis], and (ii) a kernel of size O(k 3 ). We improve both these results for SUBSET-FVS on split graphs; we derive (i) a kernel of size O(k 2 ) which is the best possible unless NP ⊆ coNP/poly, and (ii) an algorithm which solves the problem in time O * (2 k ). Our algorithm, in fact, solves SUBSET-FVS on the more general class of chordal graphs, also in O * (2 k ) time. 6 The O () notation hides polynomial factors.

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