Non-maximal decidable structures

Given any infinite structure $\mathfrak M$ with a decidable first-order theory, we give a sufficient condition in terms of the Gaifman graph of $\mathfrak M$, which ensures that $\mathfrak M$ can be expanded with some non-definable predicate in such a way that the first-order theory of the expansion is still decidable. Bibl. – 10 titles.