In this article, we consider problems on graphs of the following form: given a graph, remove p edges/vertices to achieve some property. The first kind of problems are separation problems on undirected graphs, where we aim at separating distinguished vertices in an graph. The second kind of problems are feedback set problems on group-labelled graphs, where we aim at breaking nonnull cycles in a group-labelled graph. We obtain new FPT algorithms for these different problems. A building stone for our algorithms is a general O ^*(4 ^p ) algorithm for a class of problems aiming at breaking a set of paths in a graph, provided that the set of paths has a special property called homogeneity.