  We study the recently introduced Connected Feedback Vertex Set (CFVS) problem
from the view-point of parameterized algorithms. CFVS is the connected variant
of the classical Feedback Vertex Set problem and is defined as follows: given a
graph G=(V,E) and an integer k, decide whether there exists a subset F of V, of
size at most k, such that G[V F] is a forest and G[F] is connected. We show
that 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 subroutine, a parameterized algorithm for Group Steiner Tree, a well
studied variant of Steiner Tree. We find the algorithm for Group Steiner Tree
of independent interest and believe that it will be useful for obtaining
parameterized algorithms for other connectivity problems.
