Directed figure codes are decidable

Two-dimensional structures of various kinds can be viewed as generalizations of words. Codicity verification and the defect effect, important properties related to word codes, are studied also in this context. Unfortunately, both are lost in the case of two common structures, polyominoes and figures. We consider directed figures defined as labelled polyominoes with designated start and end points, equipped with catenation operation that uses a merging function to resolve possible conflicts. We prove that in this setting verification whether a given finite set of directed figures is a code is decidable and we give a constructive algorithm. We also clarify the status of the defect effect for directed figures.