The Enumeration of Non-Isomorphic 2-Connected Planar Maps
Valery A. Liskovets, T. R. S. Walsh · Canadian Journal of Mathematics · 1983
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 ].