Our goal in this paper is to significantly decrease the compiled size of a given Boolean instance with a large representation, while preserving as much information about the instance as possible. We achieve this by assigning values to a subset of the variables in such a way that the resulting instance has a much smaller representation than the original one, and its number of solutions is almost as high as the starting one. We call the set of variable instantiations that we make the selective backbone of the solutions that we keep. Large selective backbones allow for smaller representations, but also eliminate more solutions. We compare different methods of computing the selective backbone that offer the best compromise.