Résumé
The $k$ -restricted edge-connectivity of a graph$G$ , denoted by$\lambda_k(G)$ , is defined as the minimum size of an edge set whose removal leaves exactly two connected components each containing at least$k$vertices. This graph invariant, which can be seen as a generalization of a minimum edge-cut, has been extensively studied from a combinatorial point of view. However, very little is known about the complexity of computing$\lambda_k(G)$ . Very recently, in the parameterized complexity community the notion of good edge separation of a graph has been defined, which happens to be essentially the same as the$k$ -restricted edge-connectivity. Motivated by the relevance of this invariant from both combinatorial and algorithmic points of view, in this article we initiate a systematic study of its computational complexity, with special emphasis on its parameterized complexity for several choices of the parameters. We provide a number of NP-hardness and W[1]-hardness results, as well as FPT-algorithms.