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Cold start is an essential and persistent problem in recommender systems. State-of-the-art solutions rely on training hybrid models for both cold-start and existing users/items, based on the auxiliary information. Such a hybrid model would compromise the performance of existing users/items, which might make these solutions not applicable in real-worlds recommender systems where the experience of existing users/items must be guaranteed. Meanwhile, graph neural networks (GNNs) have been demonstrated to perform effectively warm (non-cold-start) recommendations. However, they have never been applied to handle the cold-start problem in a user-item bipartite graph. This is a challenging but rewarding task since cold-start users/items do not have links. Besides, it is nontrivial to design an appropriate GNN to conduct cold-start recommendations while maintaining the performance for existing users/items. To bridge the gap, we propose a tailored GNN-based framework (GPatch) that contains two separate but correlated components. First, an efficient GNN architecture -- GWarmer, is designed to model the warm users/items. Second, we construct correlated Patching Networks to simulate and patch GWarmer by conducting cold-start recommendations. Experiments on benchmark and large-scale commercial datasets demonstrate that GPatch is significantly superior in providing recommendations for both existing and cold-start users/items.