In this paper, we investigate reducts preserving a structure induced from dominance-based rough sets as well as a structure from variable-precision dominance-based rough sets. It is shown that three kinds of reducts are obtained in the dominance-based rough set approach. For each kind of reduct, a discernibility matrix is given for the enumeration of the reducts. Moreover, it is shown that four kinds of reducts are obtained in the variable-precision dominance-based rough set approach. A counter-example showing the impossibility of reduct enumeration in the case of variable-precision dominance-based rough set approach.