  The non-emptiness, called the Domino Problem, and the characterization of the
possible entropies of $\mathbb{Z}^2$-subshifts of finite type are standard
problems of symbolic dynamics. In this article we study these questions with
horizontal constraints fixed beforehand as a parameter. We determine for which
horizontal constraints the Domino Problem is undecidable and when all
right-recursively enumerable numbers can be obtained as entropy, with two
approaches: either the additional local rules added to the horizontal
constraints can be of any shape, or they can only be vertical rules.
