We consider a class of l0-minimization problems, which is to search for the partial sparsest solution to an underdetermined linear system with additional constraints. We introduce several concepts, including lp-induced norm (0 < p < 1), maximal scaled spark and scaled mutual coherence, to develop several new uniqueness conditions for the partial sparsest solution to this class of l0-minimization problems. A further improvement of some of these uniqueness criteria have been also achieved through the so-called concepts such as maximal scaled (sub)coherence rank.