Here is another generalization of planar graphs.
Start with a d-dimensional polytope P with n vertices. For every 2-dimensional face F triangulate F by non crossing diagonals. So if F has k sides you add (k-3) edges. It is known that the total number of edges you get (including the original edges of the polytope) is at least $dn - {{d+1} \choose {2}$. 2}}$. A polytope is called "elementary" if equality holds.
We can consider the following classes of graphs:
1) E_d = Graphs of elementary d-polytopes and all their subgraphs
2) F_d = Graphs obtained by elementary d-polytopes by triangulating all 2 faces by non crossing diagonals, and all their subgraphs.
For d=3 both classes are the class of planar graphs.
Some properties of planar graphs are known or conjectured to extend.
1) (robustness; conjectured) We can start instead of polytopes by arbitrary polyhedral (d-1)-dimensional pseudomanifolds. But it is conjectured that we will get precisely the same class of graphs.
2) (duality; known) If P is elementary so is its dual P*
3) (coloring; conjectured) Graphs in E_d (and perhaps even in F_d) are (d+1)-colorable.
