Abstract: Conversational Question Generation (CQG) is a critical task for machines to assist humans in fulfilling their information needs through conversations. The task is generally cast into two different settings: answer-aware and answer-unaware. While the former facilitates the models by exposing the expected answer, the latter is more realistic and receiving growing attentions recently. What-to-ask and how-to-ask are the two main challenges in the answer-unaware setting. To address the first challenge, existing methods mainly select sequential sentences in context as the rationales. We argue that the conversation generated using such naive heuristics may not be natural enough as in reality, the interlocutors often talk about the relevant contents that are not necessarily sequential in context. Additionally, previous methods decide the type of question to be generated (boolean/span-based) implicitly. Modeling the question type explicitly is crucial as the answer, which hints the models to generate a boolean or span-based question, is unavailable. To this end, we present SG-CQG, a two-stage CQG framework. For the what-to-ask stage, a sentence is selected as the rationale from a semantic graph that we construct, and extract the answer span from it. For the how-to-ask stage, a classifier determines the target answer type of the question via two explicit control signals before generating and filtering. In addition, we propose Conv-Distinct, a novel evaluation metric for CQG, to evaluate the diversity of the generated conversation from a context. Compared with the existing answer-unaware CQG models, the proposed SG-CQG achieves state-of-the-art performance.