Detection of curvilinear structures in images has long been of interest. One of the most challenging aspects of this problem is inferring the graph representation of the curvilinear network. Most existing delineation approaches first perform binary segmentation of the image and then refine it using either a set of hand-designed heuristics or a separate classifier that assigns likelihood to paths extracted from the pixel-wise prediction. In our work, we bridge the gap between segmentation and path classification by training a deep network that performs those two tasks simultaneously. We show that this approach is beneficial because it enforces consistency across the whole processing pipeline. We apply our approach on roads and neurons datasets.