Accurate segmentation of hepatic and portal veins is critical for preoperative planning in liver surgery, especially for resection and transplantation procedures. Extensive anatomical variability, pathological alterations, and inherent class imbalance between background and vascular structures challenge this task. Current state-of-the-art deep learning approaches often fail to generalize across patient variability or maintain vascular topology, thus limiting their clinical applicability. To overcome these limitations, we propose the D-RD-UNet, a dual-stage, dual-class segmentation framework for hepatic and portal vessels. The D-RD-UNet architecture employs dense and residual connections to improve feature propagation and segmentation accuracy. Our D-RD-UNet integrates advanced data-driven preprocessing, a dual-path architecture for 3D and 4D data, with the latter concatenating computed tomography (CT) scans with four relevant vesselness filters (Sato, Frangi, OOF, and RORPO). The pipeline is completed by the first developed postprocessing multi-class vessel connectivity correction algorithm based on centerlines. Additionally, we introduce the first radius-based branching algorithm to evaluate the model's predictions locally, providing detailed insights into the accuracy of vascular reconstructions at different scales. In order to make up for the scarcity of well-annotated open datasets for hepatic vessels segmentation, we curated AIMS-HPV-385, a large, pathological, multi-class, and validated dataset on 385 CT scans. We trained different configurations of D-RD-UNet and state-of-the-art models on 327 CTs of AIMS-HPV-385. Experimental results on the remaining 58 CTs of AIMS-HPV-385 and on the 20 CTs of 3D-IRCADb-01 demonstrate superior performances of the D-RD-UNet variants over state-of-the-art methods, achieving robust generalization, preserving vascular continuity, and offering a reliable approach for liver vascular reconstructions.