Maximal planar graphs and diagonal operations

Maximal planar graphs embed in the euclidean plane as plane triangulations. Two such plane triangulations are equivalent if there is a homeomorphism of the plane which maps the vertices, edges and faces of one onto the corresponding elements of the other. A diagonal operation on a plane triangulation deletes an appropriate edge and inserts a related edge. We prove constructively that, given any two plane embeddings of maximal planar graphs of order n, which need not be isomorphic. there is a sequence of diagonal operations which transforms the fIrst into one equivalent to the second. This result was fIrst conjectured by Foulds and Robinson (1979), who made substantial progress towards its demonstration. Lehel (1980) claimed a proof, but unfortunately his argument is erroneous. Ning (1987) fIrst completed the proof with a result which fIlls the gap in the original work of Foulds and Robinson. The present paper gives a new self-contained proof, from a more geometrical viewpoint.