If any embedding of your graph in 3-space has two cycles that are linked, then your graph is intrinsically linked (such as the Petersen graph). These graphs generalise non-planar graphs since for example any flat projection of these graphs will also be non-planar.

Non-planar graphs can also be characterised by [Mac Lane's condition][1]: there is a cycle basis with a cycle sharing an edge with more than 2 cycles — i.e. the graph has a non-trivial cycle involving a non-planar edge. This is informative since it gives a more constructive explanation of non-planarity, rather than just a forbidden minors characterisation.

I would like to understand better the way that linked graphs generalise non-planar graphs. Is there a condition analogous to Mac Lane's condition that characterises linked graphs?


  [1]: https://en.wikipedia.org/wiki/Mac_Lane%27s_planarity_criterion