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The set ${\mathcal{Q}}_n$ consists of rooted planar maps with $n$ faces, every face having degree four. The set ${\mathcal{Q}}_n^{\bullet }$ consists of these quadrangulations with an additional distinguished vertex.

For the trivial bijection between planar maps and quadrangulations, start from a rooted planar map with $n$ edges. Put a new vertex in every face and join it to the original vertex at every incident corner. Delete the original edges. The two endpoints of each deleted edge and the new vertices in its two adjacent faces bound a quadrangular face, so the result lies in ${\mathcal{Q}}_n$. Its bipartition distinguishes old from new vertices and reconstructs the original map. With the standard root convention this is a bijection, and hence

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.

A rooted planar map is a planar map with a distinguished oriented edge called its root edge. Its tail is the root vertex, and the face on a chosen side of the edge may be designated as the root face.

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.