Recognition of Triangulation Duals of Simple Polygons With and Without   Holes
  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. 
Martin Derka, Alejandro L\'opez-Ortiz, Daniela Maftuleac
cs.CG