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Learning interpretable dialog structure from human-human dialogs yields basic insights into the structure of conversation, and also provides background knowledge to facilitate dialog generation. In this paper, we conduct unsupervised discovery of dialog structure from chitchat corpora, and then leverage it to facilitate dialog generation in downstream systems. To this end, we present a Discrete Variational Auto-Encoder with Graph Neural Network (DVAE-GNN), to discover a unified human-readable dialog structure. The structure is a two-layer directed graph that contains session-level semantics in the upper-layer vertices, utterance-level semantics in the lower-layer vertices, and edges among these semantic vertices. In particular, we integrate GNN into DVAE to fine-tune utterance-level semantics for more effective recognition of session-level semantic vertex. Furthermore, to alleviate the difficulty of discovering a large number of utterance-level semantics, we design a coupling mechanism that binds each utterance-level semantic vertex with a distinct phrase to provide prior semantics. Experimental results on two benchmark corpora confirm that DVAE-GNN can discover meaningful dialog structure, and the use of dialog structure graph as background knowledge can facilitate a graph grounded conversational system to conduct coherent multi-turn dialog generation.