 Trivial bijection between planar maps and quadrangulations

The trivial bijection between planar maps and quadrangulations sends a rooted planar map with $n\ge 1$ edges to a rooted planar map that is a planar quadrangulation with $n$ faces. Colour its original vertices black. Insert a white vertex in each face and connect it to every incident corner, keeping repeated incidences as separate edges. Erase the original edges. Each original edge now occupies one quadrangular face, even when its endpoints or incident faces coincide.

Fix the root convention by taking the first new outgoing edge clockwise after the original root edge at its origin. The resulting quadrangulation is a bipartite graph. Its vertex bipartition is recovered by colouring the root origin black and alternating colours along edges. Draw a black diagonal in every quadrangular face, then erase the white vertices and the quadrangulation edges. Root the recovered map at the first outgoing black diagonal counterclockwise after the quadrangulation root. These constructions are inverse.^[ref]

The bipartition is a partition of vertices, not of faces. It distinguishes the original vertices from the inserted face vertices and makes the inverse construction possible. The bijection transfers enumeration and random sampling questions between general planar maps and planar quadrangulations.