Planar quadrangulation

A planar quadrangulation is a planar map in which every face has degree four, with incidences counted with multiplicity. A pointed quadrangulation additionally has a distinguished vertex.

The trivial bijection sends a rooted planar map with $n$ edges to a rooted planar quadrangulation with $n$ faces. Put one new vertex in each face and, in each corner, connect that face vertex to the incident original vertex. Each original edge then lies inside one quadrangular face; deleting the original edges gives the quadrangulation. The face bipartition recovers the original vertices and hence the inverse map.