We present a method that detects boundaries of parts in 3D shapes represented as point clouds. Our method is based on a graph convolutional network architecture that outputs a probability for a point to lie in an area that separates two or more parts in a 3D shape. Our boundary detector is quite generic: it can be trained to localize boundaries of semantic parts or geometric primitives commonly used in 3D modeling. Our experiments demonstrate that our method can extract more accurate boundaries that are closer to ground-truth ones compared to alternatives. We also demonstrate an application of our network to fine-grained semantic shape segmentation, where we also show improvements in terms of part labeling performance.