Subset Feedback Vertex Set in Chordal and Split Graphs

Abstract

In the Subset Feedback Vertex Set (Subset-FVS) problem the input consists of 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 FPT algorithm which solves the problem in $\mathcal {O}^{\star } (2.076^k)$ time (The $\mathcal {O}^{\star } ()$ notation hides polynomial factors.) [Wahlström, Ph.D. Thesis], and (ii) a kernel of size $\mathcal {O}(k^3)$. We improve both these results for Subset-FVS on split graphs; we derive (i) a kernel of size $\mathcal {O}(k^2)$ which is the best possible unless , and (ii) an algorithm which solves the problem in time $\mathcal {O}^*(2^k)$. Our algorithm, in fact, solves Subset-FVS on the more general class of chordal graphs, also in $\mathcal {O}^*(2^k)$ time.