Counting rooted nonseparable unicursal planar maps

A map is unicursal if all its vertices are even-valent except two odd-valent vertices. This paper investigates the enumeration of rooted nonseparable unicursal planar maps and provides two functional equations satisfied by its generating functions with the number of nonrooted vertices, the number of inner faces (or the number of edges) and the valencies of the two odd vertices of maps as parameters.