Classification of the genus-1 rooted maps and associated functional relation (Q688679)

Rooted maps have traditionally been counted by deleting (or, dually, by contracting) the root-edge. This method was used to count rooted planar maps by the number of vertices and faces [\textit{W. T. Tutte}, On the enumeration of planar maps, Bull. Am. Math. Soc. 74, 64-74 (1968; Zbl 0157.311)], to develop an algorithm for counting rooted maps of a given genus by the number of vertices and faces [\textit{T. R. S. Walsh} and \textit{A. B. Lehman}, Counting rooted maps by genus. I, J. Comb. Theory, Ser. B 13, 192-218 (1972; Zbl 0228.05108)], and to obtain a generating function for counting rooted toroidal maps by the number of edges and the degree of the root-face [\textit{D. Arquès}, Relations fonctionnelles et dénombrement de cartes pointées sur le tore, J. Comb. Theory, Ser. B 43, 253-274 (1987; Zbl 0628.05040)]. This method counts rooted maps whose root-edge is an isthmus and classifies them according to whether its removal disconnects the map; to further classify rooted maps whose root- edge is not an isthmus, another reduction---contracting the root-face to a point---was invented and applied to planar maps in [\textit{D. Arquès}, Une relation fonctionnelle nouvelle sur les cartes planaires pointées, J. Comb. Theory, Ser. B 39, 27-42 (1985; Zbl 0571.05001)]. Here it is applied to rooted toroidal maps to classify those whose root-edge is not an isthmus into six classes depending upon various properties of the connected components into which the map is decomposed by deleting the bounding edges of the root-face, and to find a generating function for counting the rooted toroidal maps in each class by the number of edges and the degree of the root-face.