Fragmentation has been recently proposed as a promising approach to protect the confidentiality of sensitive associations whenever data need to undergo external release or storage. By splitting attributes among different fragments, fragmentation guarantees confidentiality of the associations among these attributes under the assumption that such associations cannot be reconstructed by re-combining the fragments. We note that the requirement that fragments do not have attributes in common, imposed by previous proposals, is only a necessary, but not sufficient, condition to ensure that information in different fragments cannot be recombined as dependencies may exist among data enabling some form of linkability. In this paper, we identify the problem of improper information leakage due to data dependencies, provide a formulation of the problem based on anatural graphical modeling, and present an approach to tackle it in an efficient and scalable way.