Computable graphs of finite $\Delta_\alpha^0$-dimensions

In present article, we prove the following assertions:

• For every computable successor ordinal $\alpha$, there exists a $\Delta_\alpha^0$-categorical directed graph (symmetric, irreflexive graph) which is not relatively $\Delta_\alpha^0$-categorical, i.e. no formally $\Sigma_\alpha^0$-Scott family exists for such a structure.
• For every computable successor ordinal $\alpha$, there exists an intrinsically $\Sigma_\alpha^0$-relation on universe of a computable directed graph (symmetric, irreflexive graph which is not a relatively intrinsically $\Sigma_\alpha^0$-relation.
• For every computable successor ordinal $\alpha$ and finite $n$, there exists a $\Delta_\alpha^0$-categorical directed graph (symmetric, irreflexive graph) whose $\Delta_\alpha^0$-dimension is equal to $n$.
• For every computable successor ordinal $\alpha$, there exists a directed graph (symmetric, irreflexive graph) possesing presentations only in the degrees of sets $X$ such that $\Delta_\alpha^0(X)\ne\Delta_\alpha^0$. In particular, for each finite $n$, there exist is a structure with presentations in just the non-low $n$ degrees.