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A novel graph-cuts-based method is proposed for reconstructing open surfaces from unordered point sets. Through a Boolean operation on the crust around the data set, the open surface problem is translated to a watertight surface problem within a restricted region. Integrating the variational model, Delaunay-based tetrahedral mesh and multiphase technique, the proposed method can reconstruct open surfaces robustly and effectively. Furthermore, a surface reconstruction method with domain decomposition is presented, which is based on the new open surface reconstruction method. This method can handle more general surfaces, such as nonorientable surfaces. The algorithm is designed in a parallel-friendly way and necessary measures are taken to eliminate cracks and conflicts between the subdomains. Numerical examples are included to demonstrate the robustness and effectiveness of the proposed method on watertight, open orientable, open nonorientable surfaces and combinations of such.