Counting rooted maps by genus III: Nonseparable maps

A map on an orientable surface is called separable if its underlying graph can be disconnected by splitting a vertex into two pieces, each containing a positive number of edge-ends consecutively ordered with respect to counter-clockwise rotation around the original vertex. This definition is shown to be equivalent for planar maps to Tutte's definition of a separable planar map. We develop a procedure for determining the generating functions N_g,b(x) = Σ_p=0^∞n_g,b,px^p, where n_g,b,p is the number of rooted nonseparable maps with b + p edges and p + 1 vertices on an orientable surface of genus g. Similar results are found for tree-rooted maps.