A FUNCTIONAL EQUATION FOR ENUMERATING NONSEPARABLE PLANAR MAPS WITH VERTEX PARTITION

A map is a 2-cell embedding of a graph on the surface. If the surface is the plane, or the sphere, the map is called a planar map. A map is said to be rooted, if one of its edges is marked with a direction chosen beforchand. The marked edge is called the root-edge; its tail vertex, the root-vertex; the face on the right-hand side of the root-edge, the root-face. Without loss of generality, the root-face may be chosen as the infinite face.