On counting planar embeddings

A method for counting the embeddings of a connected but not necessarily biconnected planar graph is given. The method relates the embedding of edges around an articulation point to a tree structure called an embedding tree. MacLane [S] gives a method for counting the number of distinct embeddings of a biconnected planar graph. This method is based on a decomposition into triconnetted components and the fact that each triconnected planar graph has a unique embedding, up to reflection. More recently linear time algorithms have been proposed for counting the number of embeddings and building efficient data structures that represent the set of all embeddings (so that these can be enumerated, or additional constraints can be incorporated). Chiba et al. [3] represent all embeddings using PQ-trees; their algorithm is based on the planarity testing algorithm of Booth and Luecker [l]. Another approach, using a variant of the Hopcroft-Tarjan algorithm [6], is given by Cai et al. [2]. The latter also gives a formula for counting embeddings in a not necessarily biconnected graph. The purpose of this note is to give an alternate method for counting the number of embeddings of a connected planar graph that is not necessarily biconnected. The method presented here is interesting because it relates each edge sequence around an articulation point to a special tree called an embedding tree and then uses methods for counting trees. Like that of Cai et al., our method yields a linear time algorithm for counting the number of embeddings of an arbitrary graph or for enumerating those Correspondence to: Matthias F.M. Stallmann, Department of Computer Science, North Carolina State University, Raleigh, NC 27695-8206, USA. 0012-365X/93/$06.00