A few reasons for the importance of planarity having little to do with the need for maps:

• A matroid is both graphic and co-graphic if and only if it is the graphic matroid of a planar graph

• Planar graphs are the graphs with Colin de Verdiere invariant ≤ 3. As such they form a sequence with the trees, outerplanar graphs, planar graphs, and linklessly embeddable graphs.

• The graphs of three-dimensional convex polyhedra are exactly the 3-connected planar graphs (Steinitz's theorem).

• A minor-closed graph family has bounded treewidth if and only if it does not include all the planar graphs.