= Trivial bijection between planar maps and quadrangulations
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.
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