Enumeration of Eulerian planar near-quadrangulations

A map is called Eulerian if all the valencies of its vertices are even. If a map has all but one of its faces quadragles, then it is called a near-quadrangulation. Up until now, there is no research work on Eulerian planar near-quadrangulations. The main result in this paper is to establish a form of functional equation satisfied by enufunction of rooted Eulerian planar near-quadrangulations with the valency of the root-face and the number of non-rooted faces as parameters.