Redactable signatures for linear-structured data such as strings have already been studied in the literature. In this paper, we propose a formal security model for leakage-free redactable signatures (LFRS) that is general enough to address authentication of not only trees but also graphs and forests. LFRS schemes have several applications, especially in enabling secure data management in the emerging cloud computing paradigm as well as in healthcare, finance and biological applications. We have also formally defined the notion of secure names. Such secure names facilitate leakage-free verification of ordering between siblings/nodes. The paper also proposes a construction for secure names, and a construction for leakagefree redactable signatures based on the secure naming scheme. The proposed construction computes a linear number of signatures with respect to the size of the data object, and outputs only one signature that is stored, transmitted and used for authentication of any tree, graph and forest.