Various kinds of decipherability of codes, weaker than unique decipherability, have been studied since mid-1980s. We consider decipherability of directed gure codes, where directed gures are de ned as labelled polyomi- noes with designated start and end points, equipped with catenation operation that may use a merging function to resolve possible con icts. This setting ex- tends decipherability questions from words to 2D structures. In the present paper we develop a (variant of) domino graph that will allow us to decide some of the decipherability kinds by searching the graph for speci c paths. Thus the main result characterizes directed gure decipherability by graph properties.