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The recent advancements of three-dimensional (3D) data acquisition devices have spurred a new breed of applications that rely on point cloud data processing. However, processing a large volume of point cloud data brings a significant workload on resource-constrained mobile devices, prohibiting from unleashing their full potentials. Built upon the emerging paradigm of device-edge co-inference, where an edge device extracts and transmits the intermediate feature to an edge server for further processing, we propose Branchy-GNN for efficient graph neural network (GNN) based point cloud processing by leveraging edge computing platforms. In order to reduce the on-device computational cost, the Branchy-GNN adds branch networks for early exiting. Besides, it employs learning-based joint source-channel coding (JSCC) for the intermediate feature compression to reduce the communication overhead. Our experimental results demonstrate that the proposed Branchy-GNN secures a significant latency reduction compared with several benchmark methods.