  We investigate the problem of determining if a given graph corresponds to the
dual of a triangulation of a simple polygon. This is a graph recognition
problem, where in our particular case we wish to recognize a graph which
corresponds to the dual of a triangulation of a simple polygon with or without
holes and interior points. We show that the difficulty of this problem depends
critically on the amount of information given and we give a sharp boundary
between the various tractable and intractable versions of the problem.
