Identifying and segmenting hepatic and portal veins (two predominant vascular systems in the liver, from CT scans) play a crucial role for clinicians in preoperative planning for treatment strategies. However, existing segmentation models often struggle to capture fine details of minor veins. In this article, we introduce Vessel Structure-aware Network (VSNet), a multi-task learning model with vessel-growing decoder, to address the challenge. VSNet excels at accurate segmentation by capturing the topological features of both minor veins while preserving correct connectivity from minor vessels to trucks. We also build and publish the largest dataset (303 cases) for hepatic and portal vessel segmentation. Through comprehensive experiments, we demonstrate that VSNet achieves the best Dice for hepatic vein of 0.824 and portal vein of 0.807 on our proposed dataset, significantly outperforming other popular segmentation models. The source code and dataset are publicly available at https://github.com/XXYZB/VSNet.