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With the development of deep convolutional neural networks, medical image segmentation has achieved a series of breakthroughs in recent years. However, the high-performance convolutional neural networks always mean numerous parameters and high computation costs, which will hinder the applications in clinical scenarios. Meanwhile, the scarceness of large-scale annotated medical image datasets further impedes the application of high-performance networks. To tackle these problems, we propose Graph Flow, a comprehensive knowledge distillation framework, for both network-efficiency and annotation-efficiency medical image segmentation. Specifically, our core Graph Flow Distillation transfer the essence of cross-layer variations from a well-trained cumbersome teacher network to a non-trained compact student network. In addition, an unsupervised Paraphraser Module is integrated to purify the knowledge of the teacher network, which is also beneficial for the stabilization of training procedure. Furthermore, we build a unified distillation framework by integrating the adversarial distillation and the vanilla logits distillation, which can further refine the final predictions of the compact network. With different teacher networks (conventional convolutional architecture or prevalent transformer architecture) and student networks, we conduct extensive experiments on four medical image datasets with different modalities (Gastric Cancer, Synapse, BUSI, and CVC-ClinicDB).We demonstrate the prominent ability of our method which achieves competitive performance on these datasets. Moreover, we demonstrate the effectiveness of our Graph Flow through a novel semi-supervised paradigm for dual efficient medical image segmentation. Our code will be available at Graph Flow.