Further to the similarities between numbering atoms in a structure, and generating unique graphs here is this:

which shows the same molecule with two different numberings on the left, and the resulting graphs on the right. The double bond is not shown on the graphs; but it would probably have to be a labelling of the edge, rather than an actual multiple edge, to still be a simple graph.

So, this quickly shows how - if you start with the vertices of the graph and connect 'all possible ways' - you get molecules that are isomorphic, but numbered differently. Therefore (perhaps) the numbering of the vertices and edges is one of he keys to not creating all the isomorphs and then having to expensively check them all.

## Comments

How does that relate to your above analysis?

It could really be any mechanism, but there should be some rule to say "try only these possible bonds" rather than a method to find out which ones turn out to be the wrong ones.