We consider several kinds of decipherability of directed figure codes, where directed figures are defined as labelled polyominoes with designated start and end points, equipped with catenation that may use a merging function for overlapping regions. This setting extends decipherability questions from words to 2D structures. In this paper we develop a (variant of) domino graph that will allow us to decide some of the decipherability kinds by searching the graph for specific paths. Thus the main result characterizes directed figure decipherability by graph properties.
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