Graph-processing systems, including Graph Database Management Systems (GDBMSes) and graph libraries, are designed to analyze and manage graph data efficiently. They are widely used in applications such as social networks, recommendation systems, and fraud detection. However, logic bugs in these systems can lead to incorrect results, compromising the reliability of applications. While recent research has explored testing techniques specialized for GDBMSes, it is unclear how to adapt them to graph-processing systems in general. This paper proposes Graph-cutting, a universal approach for detecting logic bugs in both GDBMSes and various algorithms in graph libraries. Our key idea is inspired by the observation that certain graph patterns are critical for various graph-processing tasks. Dividing graph data into subgraphs that preserve those patterns establishes a natural relationship between query results on the original graph and its subgraphs, allowing for the detection of logic bugs when this relationship is violated. We implemented Graph-cutting as a tool, GSlicer, and evaluated it on 3 popular graph-processing systems, NetworkX, Neo4j, and Kùzu. GSlicer detected 39 unique and previously unknown bugs, out of which 34 have been fixed and confirmed by developers. At least 8 logic bugs detected by GSlicer cannot be detected by baseline strategies. Additionally, by leveraging just a few concrete relationships, Graph-cutting can cover over 100 APIs in NetworkX. We expect this technique to be widely applicable and that it can be used to improve the quality of graph-processing systems broadly.