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Given the convenience of collecting information through online services, recommender systems now consume large scale data and play a more important role in improving user experience. With the recent emergence of Graph Neural Networks (GNNs), GNN-based recommender models have shown the advantage of modeling the recommender system as a user-item bipartite graph to learn representations of users and items. However, such models are expensive to train and difficult to perform frequent updates to provide the most up-to-date recommendations. In this work, we propose to update GNN-based recommender models incrementally so that the computation time can be greatly reduced and models can be updated more frequently. We develop a Graph Structure Aware Incremental Learning framework, GraphSAIL, to address the commonly experienced catastrophic forgetting problem that occurs when training a model in an incremental fashion. Our approach preserves a user's long-term preference (or an item's long-term property) during incremental model updating. GraphSAIL implements a graph structure preservation strategy which explicitly preserves each node's local structure, global structure, and self-information, respectively. We argue that our incremental training framework is the first attempt tailored for GNN based recommender systems and demonstrate its improvement compared to other incremental learning techniques on two public datasets. We further verify the effectiveness of our framework on a large-scale industrial dataset.