Degrees of Selector Functions and Relative Computable Categoricity

We study the classes of Turing degrees of selector functions in which a rigid computable structure is relatively computably categorical. It is proved that for some structures such classes of degrees can be represented as the unions of upper cones of c.e. degrees. In addition we show that there are non-c.e. upper cones realized as the degrees in which some computable structure is relatively computably categorical.