We have the definition of planar graph, which captures the idea of all the connections "lying in a plane." Because any graph can be embedded in three-dimensional space without crossing, this definition does not generalize. However, there is a definition of a graph "lying in a volume-space" which is the definition of a linklessly embeddable graph.
So we have:
1-dimension :: path graph (embeddable in a line)
2-dimension :: planar graph (embeddable in a plane)
3-dimension :: linklessly embeddable graph (embeddable in space)
What are the generalizations for $n$ dimensions?
Best Answer
Yes. The generalization is the Colin de Verdière graph invariant ($\mu$). We have the following cases.
The "$\leq$" is only necessary because any outerplanar graph is a planar graph, and is a linklessly embeddable graph, and so on. If we set $\mu = 3$ we get the strict planar graphs, excluding outerplanar graphs and path graphs (or their disjoint unions).