Many state-of-the-art delineation methods rely on supervised machine learning algorithms. As a result, they require manually annotated training data, which is tedious to obtain. Furthermore, even minor classification errors may significantly affect the topology of the final result. In this paper we propose a generic approach to addressing both of these problems by taking into account the influence of a potential misclassification on the resulting delineation. In an Active Learning context, we identify parts of linear structures that should be annotated first in order to train a classifier effectively. In a proofreading context, we similarly find regions of the resulting reconstruction that should be verified in priority to obtain a nearly-perfect result. In both cases, by focusing the attention of the human expert on potential classification mistakes which are the most critical parts of the delineation, we reduce the amount of required supervision. We demonstrate the effectiveness of our approach on microscopy images depicting blood vessels and neurons.