The code problem for directed figures

TL;DR: It is shown that depending on the catenation type the question whether a given set of directed figures is a code is decidable or not, and a constructive proof is given which leads to a straightforward algorithm.

Abstract: We consider directed figures defined as labelled polyominoes with designated start and end points, with two types of catenation operations. We are especially interested in codicity verification for sets of figures, and we show that depending on the catenation type the question whether a given set of directed figures is a code is decidable or not. In the former case we give a constructive proof which leads to a straightforward algorithm.