Exact algorithms for restricted subset feedback vertex set in chordal and split graphs

The RESTRICTED SUBSET FEEDBACK VERTEX SET problem (R-SFVS) takes a graph G = (V, E), a terminal set T C V, an integer k as the input. The task is to determine whether there exists a subset S C V ⧵ T of at most k vertices, after deleting which no terminal in T is contained in a cycle in the remaining graph. R-SFVS is NP-complete even when the input graph is restricted to split graphs. In this paper, we mainly show that R-SFVS in chordal , split graphs can be solved in 0(1.1550|V|) time and exponential space or in 0(1.1605|V|) time and polynomial space, significantly improving all previous results. As a by-product, we show that the MAXIMUM INDEPENDENT SET problem parameterized by the edge clique cover number is fixed-parameter tractable. Furthermore, by using a simple reduction from R-SFVS to VERTEX COVER, we obtain an 0*(1.2738k)-time parameterized algorithm and a tight 0(k2)-kernel for R-SFVS in chordal and split graphs.