  The maximum independent sets in the Doob graphs D(m,n) are analogs of the
distance-2 MDS codes in Hamming graphs and of the latin hypercubes. We prove
the characterization of these sets stating that every such set is semilinear or
reducible. As related objects, we study vertex sets with maximum cut (edge
boundary) in D(m,n) and prove some facts on their structure. We show that the
considered two classes (the maximum independent sets and the maximum-cut sets)
can be defined as classes of completely regular sets with specified 2-by-2
quotient matrices. It is notable that for a set from the considered classes,
the eigenvalues of the quotient matrix are the maximum and the minimum
eigenvalues of the graph. For D(m,0), we show the existence of a third,
intermediate, class of completely regular sets with the same property.
