Do you know anything special about that kind of planar graphs? An article that covers these graphs might be helpful. 



Planar Graphs with #Vertices = #Faces
Unlike some other planar graphs, these ones always contain at least one triangular face and at least one vertex of degree $\le 3$. The reason for the first property is that, by Euler's formula, the number of edges in a planar graph with no triangles is at most $2n-4$, but your graphs always have exactly $2n-2$ edges. The second property is just the dual of the first.


As Douglas Zare already said, I doubt they have any more interesting properties than that, though. They can be made by mixing parts of graphs that are dense and other parts that are sparse until you have the right overall balance of edges and faces, so near any particular vertex they can look like any other planar graph.


