Registration is a basic yet crucial task in point cloud processing. In correspondence-based point cloud registration, matching correspondences by point feature techniques may lead to an extremely high outlier (false correspondence) ratio. Current outlier removal methods still suffer from low efficiency, accuracy, and recall rate. We use an intuitive method to describe the 6-DOF (degree of freedom) curtailment process in point cloud registration and propose an outlier removal strategy based on the reliability of the correspondence graph. The method constructs the corresponding graph according to the given correspondences and designs the concept of the reliability degree of the graph node for optimal candidate selection and the reliability degree of the graph edge to obtain the global maximum consensus set. The presented method achieves fast and accurate outliers removal along with gradual aligning parameters estimation. Extensive experiments on simulations and challenging real-world datasets demonstrate that the proposed method can still perform effective point cloud registration even the correspondence outlier ratio is over 99%, and the efficiency is better than the state-of-the-art. Code is available at https://github.com/WPC-WHU/GROR.