Subset Feedback Vertex Set in Chordal and Split Graphs
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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 -Complete even in split graphs. Cygan et al. proved that Subset-FVS is fixed parameter tractable () 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-FVSrestricted to split graphs, (i) an algorithm which solves the problem in (2.076^k) time [The () notation hides polynomial factors.]Subset-FVS in Chordal 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 ⊆/ 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.
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