Many problems in computer vision and biomedical image analysis benefit from representing an image as a set of superpixels or supervoxels. Inspired by this, we propose to partition a cortical surface into a collection of small patches, namely supervertices, with quasi-uniform size and coverage, compactness, and also smooth boundaries that align sulcal fundi or gyral crest curves on cortical surfaces. The ultimate goal of supervertices partition of the cortical surfaces is to use supervertices as primitives for cortical surface analysis, such as the extraction of sulcal fundi or gyral crest curves by linking boundaries of supervertices, and also the parcellation of cortical surfaces by labeling supervertices instead of vertices. We formulate the supervertices partition as an energy minimization problem and optimize it with graph cuts. Specifically, our energy function encourages the supervertices with compact shapes and smooth boundaries at flat cortical regions and also the supervertices with boundaries aligned with the sulcal fundi or gyral crest curves at highly bended cortical regions. The method has been successfully applied to cortical surfaces of brain MR images in NAMIC and MSC datasets. Both qualitative and quantitative evaluation results demonstrate its validity. We also show an application, i.e., extraction of gyral crest curves on cortical surfaces by linking boundaries of supervertices.