User-provided object bounding box is a simple and popular interaction paradigm considered by many existing interactive image segmentation frameworks. However, these frameworks tend to exploit the provided bounding box merely to exclude its exterior from consideration and sometimes to initialize the energy minimization. In this paper, we discuss how the bounding box can be further used to impose a powerful topological prior, which prevents the solution from excessive shrinking and ensures that the user-provided box bounds the segmentation in a sufficiently tight way. The prior is expressed using hard constraints incorporated into the global energy minimization framework leading to an NP-hard integer program. We then investigate the possible optimization strategies including linear relaxation as well as a new graph cut algorithm called pinpointing. The latter can be used either as a rounding method for the fractional LP solution, which is provably better than thresholding-based rounding, or as a fast standalone heuristic. We evaluate the proposed algorithms on a publicly available dataset, and demonstrate the practical benefits of the new prior both qualitatively and quantitatively.