Accurate segmentation of liver vessels is crucial for medical applications due to its pivotal role in diagnosing liver diseases, planning surgical interventions, and assessing treatment effectiveness. In this paper, we present a new dataset for liver vessel separation and propose two novel plug-and-play modules integrated into deep learning frameworks for liver vessel segmentation. The first module, termed as the skeleton tracking module, addresses the issue of segmentation fragmentation by effectively tracking the vessel skeletons. The second module, the graph attention module, is introduced for vessel separation. We demonstrate the effectiveness of our proposed approach through comprehensive experiments, showcasing significant improvements in segmentation accuracy. The dataset is publicly available, fostering research and development. https://github.com/oneway-phil/SKTS-GAT/tree/main .