Exact and Parameterized Algorithms for Restricted Subset Feedback Vertex Set in Chordal Graphs

The Restricted Subset Feedback Vertex Set problem (R-SFVS) takes a graph G = (V, E), a terminal set T subset of V, and an integer k as the input. The task is to determine whether there exists a subset S subset of 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 chordal graphs. In this paper, we show that R-SFVS in chordal graphs can be solved in time O(1.1550 vertical bar V vertical bar), significantly improving all the previous results. As a by-product, we prove 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 a 1.2738(k)vertical bar V vertical bar(O(1))-time parameterized algorithm and an O(k(2))-kernel for R-SFVS in chordal graphs.