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Graph Neural Networks (GNNs) are popular machine learning methods for modeling graph data. A lot of GNNs perform well on homophily graphs while having unsatisfactory performance on heterophily graphs. Recently, some researchers turn their attention to designing GNNs for heterophily graphs by adjusting the message passing mechanism or enlarging the receptive field of the message passing. Different from existing works that mitigate the issues of heterophily from model design perspective, we propose to study heterophily graphs from an orthogonal perspective by rewiring the graph structure to reduce heterophily and making the traditional GNNs perform better. Through comprehensive empirical studies and analysis, we verify the potential of the rewiring methods. To fully exploit its potential, we propose a method named Deep Heterophily Graph Rewiring (DHGR) to rewire graphs by adding homophilic edges and pruning heterophilic edges. The detailed way of rewiring is determined by comparing the similarity of label/feature-distribution of node neighbors. Besides, we design a scalable implementation for DHGR to guarantee high efficiency. DHRG can be easily used as a plug-in module, i.e., a graph pre-processing step, for any GNNs, including both GNN for homophily and heterophily, to boost their performance on the node classification task. To the best of our knowledge, it is the first work studying graph rewiring for heterophily graphs. Extensive experiments on 11 public graph datasets demonstrate the superiority of our proposed methods.