We present two new combinatorial tools for the design of parameterized algorithms. The first is a simple linear time randomized algorithm that given as input a d-degenerate graph G and an integer k, outputs an independent set Y, such that for every independent set X in G of size at most k, the probability that X is a subset of Y is at least (((d+1)kk) . k(d+1))-1. The second is a new (deterministic) polynomial time graph sparsification procedure that given a graph G, a set T = s1, t1 , s2, t2, .... , s , t of terminal pairs, and an integer k, returns an induced subgraph G∗ of G that maintains all the inclusion minimal multicuts of G of size at most k and does not contain any (k+2)-vertex connected set of size 2O(k). In particular, G∗ excludes a clique of size 2O(k) as a topological minor. Put together, our new tools yield new randomized fixed parameter tractable (FPT) algorithms for STABLE s-t SEPARATOR, STABLE ODD CYCLE TRANSVERSAL, and STABLE MULTICUT on general graphs, and for STABLE DIRECTED FEEDBACK VERTEX SET on d-degenerate graphs, resolving two problems left open by Marx et al. [ACM Transactions on Algorithms, 2013{. All of our algorithms can be derandomized at the cost of a small overhead in the running time. © 2020 ACM.}, author_keywords={Independece covering family; parameterized algorithms; stable multicut; stable OCT; stable s-t separator}, keywords={Graph algorithms; Parameter estimation; Polynomial approximation; Separators, Combinatorial tools; Feedback vertex set; Graph sparsification; Induced subgraphs; Odd cycle transversals; Parameterized algorithm; Randomized Algorithms; Topological-minor, Graph theory}, publisher={Association for Computing Machinery}, issn={15496325}, language={English}, abbrev_source_title={ACM Trans. Algorithms}, document_type={Conference Paper}, source={Scopus},