The 3D visibility skeleton is a data structure that encodes the global visibility information of a set of 3D objects. While it is useful in answering global visibility queries, its large size often limits its practical use. In this paper, we address this issue by proposing a subset of the visibility skeleton, which is about $25\%$ to $50\%$ of the whole set. We show that the rest of the data structure can be recovered from the subset as needed, partially or completely. Our recovery method is efficient in the sense that it is output-dependent. We also prove that this subset is minimal for the complexity of our recovery method.