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In recent years, graph neural networks (GNN) have achieved significant developments in a variety of graph analytical tasks. Nevertheless, GNN's superior performance will suffer from serious damage when the collected node features or structure relationships are partially missing owning to numerous unpredictable factors. Recently emerged graph completion learning (GCL) has received increasing attention, which aims to reconstruct the missing node features or structure relationships under the guidance of a specifically supervised task. Although these proposed GCL methods have made great success, they still exist the following problems: the reliance on labels, the bias of the reconstructed node features and structure relationships. Besides, the generalization ability of the existing GCL still faces a huge challenge when both collected node features and structure relationships are partially missing at the same time. To solve the above issues, we propose a more general GCL framework with the aid of self-supervised learning for improving the task performance of the existing GNN variants on graphs with features and structure missing, termed unsupervised GCL (UGCL). Specifically, to avoid the mismatch between missing node features and structure during the message-passing process of GNN, we separate the feature reconstruction and structure reconstruction and design its personalized model in turn. Then, a dual contrastive loss on the structure level and feature level is introduced to maximize the mutual information of node representations from feature reconstructing and structure reconstructing paths for providing more supervision signals. Finally, the reconstructed node features and structure can be applied to the downstream node classification task. Extensive experiments on eight datasets, three GNN variants and five missing rates demonstrate the effectiveness of our proposed method., Comment: Accepted by 23rd IEEE International Conference on Data Mining (ICDM 2023)