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Test-time adaptation aims to adapt a well-trained model using test data only, without accessing training data. It is a crucial topic in machine learning, enabling a wide range of applications in the real world, especially when it comes to data privacy. While existing works on test-time adaptation primarily focus on Euclidean data, research on non-Euclidean graph data remains scarce. Prevalent graph neural network methods could encounter serious performance degradation in the face of test-time domain shifts. In this work, we propose a novel method named Adaptive Subgraph-based Selection and Regularized Prototype Supervision (ASSESS) for reliable test-time adaptation on graphs. Specifically, to achieve flexible selection of reliable test graphs, ASSESS adopts an adaptive selection strategy based on fine-grained individual-level subgraph mutual information. Moreover, to utilize the information from both training and test graphs, ASSESS constructs semantic prototypes from the well-trained model as prior knowledge from the unknown training graphs and optimizes the posterior given the unlabeled test graphs. We also provide a theoretical analysis of the proposed algorithm. Extensive experiments verify the effectiveness of ASSESS against various baselines.