The computation of minimal rare itemsets is a well known task in data mining, with numerous applications, e.g., drugs effects analysis and network security, among others. This paper presents a novel approach to the computation of minimal rare itemsets. First, we introduce a generalization of the traditional minimal rare itemset model called k-minimal rare itemset. A k-minimal rare itemset is defined as an itemset that becomes frequent or rare based on the removal of at least k or at most (k − 1) items from it. We claim that our work is the first to propose this generalization in the field of data mining. We then present a SAT-based framework for efficiently discovering k-minimal rare itemsets from large transaction databases. Afterwards, by partitioning the k-minimal rare itemset mining problem into smaller sub-problems, we aim to make it more manageable and easier to solve. Finally, to evaluate the effectiveness and efficiency of our approach, we conduct extensive experimental analysis using various popular datasets. We compare our method with existing specialized algorithms and CP-based algorithms commonly used for this task.