Unsupervised domain adaptation (UDA), aiming to adapt the model to an unseen domain without annotations, has drawn sustained attention in surgical instrument segmentation. Existing UDA methods neglect the domain-common knowledge of two datasets, thus failing to grasp the inter-category relationship in the target domain and leading to poor performance. To address these issues, we propose a graph-based unsupervised domain adaptation framework, named Interactive Graph Network (IGNet), to effectively adapt a model to an unlabeled new domain in surgical instrument segmentation tasks. In detail, the Domain-common Prototype Constructor (DPC) is first advanced to adaptively aggregate the feature map into domain-common prototypes using the probability mixture model, and construct a prototypical graph to interact the information among prototypes from the global perspective. In this way, DPC can grasp the co-occurrent and long-range relationship for both domains. To further narrow down the domain gap, we design a Domain-common Knowledge Incorporator (DKI) to guide the evolution of feature maps towards domain-common direction via a common-knowledge guidance graph and category-attentive graph reasoning. At last, the Cross-category Mismatch Estimator (CME) is developed to evaluate the category-level alignment from a graph perspective and assign each pixel with different adversarial weights, so as to refine the feature distribution alignment. The extensive experiments on three types of tasks demonstrate the feasibility and superiority of IGNet compared with other state-of-the-art methods. Furthermore, ablation studies verify the effectiveness of each component of IGNet. The source code is available at https://github.com/CityU-AIM-Group/Prototypical-Graph-DA.