To solve the data-flow graph partitioning problem of deep learning in distributed architecture, this paper presents a vertex degree aware two-stage graph partitioning method. At first stage, a vertex degree aware method is designed to preliminarily partition data-flow graph; at second stage, a fuzzy clustering based edge re-distributing algorithm is formulated for load balancing across computing nodes. Combining two stages, the framework of the two-stage method is constructed. Theoretical analysis demonstrates that the proposed method can achieve smaller upper bound of communication cost than comparison methods. Experimental results validate that the proposed method outperforms comparison methods in terms of replication factor, edge imbalance and execution time of data-flow graph computing.