Bisingular maps on the torus (Q874353)

A map is bisingular if each edge is either a loop or an isthmus (incident on both sides with the same face). A loop is called planar if one of the two regions into which it divides the imbedding surface is homeomorphic to a disc; otherwise is called essential. In this paper a functional equation is presented for the generating function for the number of rooted bisingular maps on the sphere with a given number of loops, a given number of ithmuses and a given root-vertex valency. A similar result is obtained for rooted bisingular maps on the torus as a function of four parameters: the number of planar loops, the number of essential loops, the number of isthmuses and the root-vertex valency. Explicit enumeration formulae are deduced in the cases where the root-vertex valency is ignored and there are no essential loops.