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 unified approach to address both of these problems by taking into account the influence of a potential misclassification on the resulting delineation. In an Active Learning context, we find parts of linear structures that should be annotated first in order to train an effective classifier. 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 the potential classification mistakes, which are the most critical parts of the delineation, we reduce the amount of annotation effort. We demonstrate the effectiveness of our approach on a variety of datasets that comprise both biomedical and aerial imagery.