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)k kí k(d + 1)-1. The second is a new (deterministic) polynomial time graph sparsification procedure that given a graph G, a set T = ffs1; t1g; fs2; t2g; : : : ; fs'; t'gg 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. © Copyright 2018 by SIAM.