{FPT} Algorithms for Connected Feedback Vertex Set

We study the recently introduced \textscConnected Feedback Vertex Set (CFVS)} problem from the view-point of parameterized algorithms. CFVS is the connected variant of the classical \textsc{Feedback Vertex Set} problem and is defined as follows: given a graph G=(V,E) and an integer k, decide whether there exists F\subseteq V, |F| ≤q k, such that G[V \setminus F] is a forest and G[F] is connected. We show that \textsc{Connected Feedback Vertex Set} can be solved in time O(2^{O(k)}n^{O(1)}) on general graphs and in time O(2^{O(\sqrt{k}\log k)}n^{O(1)}) on graphs excluding a fixed graph H as a minor. Our result on general undirected graphs uses, as a subroutine, a parameterized algorithm for \textsc{Group Steiner Tree}, a well studied variant of \textsc{Steiner Tree}. We find the algorithm for \textsc{Group Steiner Tree of independent interest and believe that it could be useful for obtaining parameterized algorithms for other connectivity problems