A planar map is a planar graph together with a fixed embedding in the sphere, considered up to orientation-preserving homeomorphism. A rooted planar map has a distinguished oriented edge, which removes most embedding symmetries.
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.
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 edges to a rooted planar quadrangulation with 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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