The Enumeration of Non-Isomorphic 2-Connected Planar Maps

Since Tutte initiated the systematic enumeration of planar maps [11], most of the literature on the subject has dealt with rooted maps (i.e., with maps whose automophism group has been trivialized by distinguishing a doubly-oriented edge). In particular, Tutte proved [11] that the number B′(n) of rooted planar 2-connected (i.e., non-separable) maps with n ≧ 1 edges is expressed by the formula Recently one of the authors developed a general technique for enumerating unrooted planar maps considered up to orientationpreserving isomorphisms (see [6] and [8]). This technique, which is based on combinatorial map theory, Burnside’s lemma [3, p. 181] and the concept of a quotient map (see Section 1.4), was used to find, with little algebraic manipulation, simple counting formulae for the numbers of non-isomorphic planar maps of several types [7].