We present a fully automatic approach for reconstructing compact 3D building models from large-scale airborne point clouds. A major challenge of urban reconstruction from airborne point clouds lies in that the vertical walls are typically missing. Based on the observation that urban buildings typically consist of planar roofs connected with vertical walls to the ground, we propose an approach to infer the vertical walls directly from the data. With the planar segments of both roofs and walls, we hypothesize the faces of the building surface, and the final model is obtained by using an extended hypothesis-and-selection-based polygonal surface reconstruction framework. Specifically, we introduce a new energy term to encourage roof preferences and two additional hard constraints into the optimization step to ensure correct topology and enhance detail recovery. Experiments on various large-scale airborne point clouds have demonstrated that the method is superior to the state-of-the-art methods in terms of reconstruction accuracy and robustness. In addition, we have generated a new dataset with our method consisting of the point clouds and 3D models of 20k real-world buildings. We believe this dataset can stimulate research in urban reconstruction from airborne point clouds and the use of 3D city models in urban applications.