Enumerating Rooted Biconnected Planar Graphs with Internally Triangulated Faces

A graph is called a triangulated planar graph if it admits a plane embedding in the plane such that all inner faces are triangle. In a rooted triangulated planar graph, a vertex and two edges incident to it are designated as an outer vertex and outer edges, respectively. Two plane embedding of rooted triangulated planar graphs are defined to be equivalent if they admit an isomorphism such that the designated vertices correspond each other. Given a positive integer n, we give an algorithm for enumerating all plane embeddings of rooted, biconnected and triangulated planar graphs with at most n vertices without delivering two equivalent embeddings. The algorithm runs in constant time per each by outputting the difference from the previous output.