This is a follow up to Harrison's question: [why planar graphs are so exceptional](https://mathoverflow.net/questions/7114/why-are-planar-graphs-so-exceptional). I would like to ask about (and collect answers to) various notions, in graph theory and beyond graph theory (topology; algebra) that generalize the notion of planar graphs, and how properties of planar graphs extend in these wider contexts.



Generalizations of Planar Graphs
I guess one possible generalization could be: an $m$-dimensional [stratified space](https://en.wikipedia.org/wiki/Topologically_stratified_space) (i.e. "manifold with singularities") which is embeddable in $2m$-dimensional Euclidean space. Every smooth manifold can be so embedded (by Whitney's theorem), but singularities may force the ambient dimension higher, as witnessed by the simple case $m=1$ in which "stratified space" is just a graph and the embeddability condition is just planarity.


(This was just invented on the spot - I have no idea if this is actually an interesting definition...)


