The accurate segmentation of blood vessels and centerline extraction are critical in vascular imaging applications, ranging from preoperative planning to hemodynamic modeling. This study introduces a novel one-stage method for simultaneous vessel segmentation and centerline extraction using a multitask neural network. We designed a hybrid architecture that integrates convolutional and graph layers, along with a task-specific loss function, to effectively capture the topological relationships between segmentation and centerline extraction, leveraging their complementary features. The proposed end-to-end framework directly predicts the centerline as a polyline with real-valued coordinates, thereby eliminating the need for post-processing steps commonly required by previous methods that infer centerlines either implicitly or without ensuring point connectivity. We evaluated our approach on a combined dataset of 142 computed tomography angiography images of the thoracic and abdominal regions from LIDC-IDRI and AMOS datasets. The results demonstrate that our method achieves superior centerline extraction performance (Surface Dice with threshold of 3 mm: 97.65%±2.07%) compared to state-of-the-art techniques, and attains the highest subvoxel resolution (Surface Dice with threshold of 1 mm: 72.52%±8.96%). In addition, we conducted a robustness analysis to evaluate the model stability under small rigid and deformable transformations of the input data, and benchmarked its robustness against the widely used VMTK toolkit.