Numbering atoms, numbering vertices

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

If I understood correctly, the trick is to detect of the numbering your started is unique. The code that the CDK deterministic generated was using, was using this approach. It would exhaustively test all possible graphs, but would stop testing a particular solution if it did not meet some uniqueness evaluation. For this it used a some normalized matrix representation of the graph, from which it started left top and scan the top left row*col to see if it matched this evaluation.

How does that relate to your above analysis?
gilleain said…
That seems like the kind of approach, yes. The trick is to know not to try a particular graph without actually generating it, let alone then testing it against example stored in memory.

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.