Point cloud segmentation is fundamental in many medical applications, such as aneurysm clipping and orthodontic planning. Recent methods mainly focus on designing powerful local feature extractors and generally overlook the segmentation around the boundaries between objects, which is extremely harmful to the clinical practice and degenerates the overall segmentation performance. To remedy this problem, we propose a GRAph-based Boundary-aware Network (GRAB-Net) with three paradigms, Graph-based Boundary-perception Module (GBM), Outer-boundary Context-assignment Module (OCM), and Inner-boundary Feature-rectification Module (IFM), for medical point cloud segmentation. Aiming to improve the segmentation performance around boundaries, GBM is designed to detect boundaries and interchange complementary information inside semantic and boundary features in the graph domain, where semantics-boundary correlations are modelled globally and informative clues are exchanged by graph reasoning. Furthermore, to reduce the context confusion that degenerates the segmentation performance outside the boundaries, OCM is proposed to construct the contextual graph, where dissimilar contexts are assigned to points of different categories guided by geometrical landmarks. In addition, we advance IFM to distinguish ambiguous features inside boundaries in a contrastive manner, where boundary-aware contrast strategies are proposed to facilitate the discriminative representation learning. Extensive experiments on two public datasets, IntrA and 3DTeethSeg, demonstrate the superiority of our method over state-of-the-art methods.