Selectively forgetting information while preserving what matters the most is becoming an increasingly important issue in many areas, including in knowledge representation and reasoning. Depending on the application at hand, forgetting operators are defined to obey different sets of desirable properties. Of the myriad of desirable properties discussed in the context of forgetting in Answer Set Programming, strong persistence, which imposes certain conditions on the correspondence between the answer sets of the program pre-and post-forgetting, and a certain independence from non-forgotten atoms, seems to best capture its essence, and be desirable in general. However, it has remained an open problem whether it is always possible to forget a set of atoms from a program while obeying strong persistence. In this paper, after showing that it is not always possible to forget a set of atoms from a program while obeying this property, we move forward and precisely characterise what can and cannot be forgotten from a program, by presenting a necessary and sufficient criterion. This characterisation allows us to draw some important conclusions regarding the existence of forgetting operators for specific classes of logic programs, to characterise the class of forgetting operators that achieve the correct result whenever forgetting is possible, and investigate the related question of determining what we can forget from some specific logic program.