Comparing Kiraly (Exhaustive) Graph Generation with nAUTy Output

So recently I was asked about Király's method for generating all graphs from a degree sequence. While refactoring some of the code that I wrote to do this, I also made some tests. Specifically, coverage tests to check that the generation was actually exhaustive. I know it's redundant, but I have good tools to remove duplicate graphs - or I thought that I did…

Here a rough flowchart of the procedure here, starting with a number ('n') that is passed to Dreadnaut (the interface to nAUTy) to generate graphs:

These graphs are grouped by degree sequence, and these degree sequences are fed into the KirályHHGenerator to reconstruct the set of graphs. I think that compare arrow is wrong, but never mind. The point is that the sets should be the same size.

They are for n=5,6,7 but not for 8. Oddly enough, however, there are more in the Király set than in the nAUTy set. The obvious conclusion would be that my duplicate detection is failing - in other words, I am failing to spot an isomorphism between two graphs. For example this pair:

However, two of my methods give different answers for this pair. The signatures method says they are different, while the partition refinement method says they are the same. Odd - and more investigation is needed before I am certain that geng has missed some graphs here...

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Adamantane, Diamantane, Twistane

After cubane, the thought occurred to look at other regular hydrocarbons. If only there was some sort of classification of chemicals that I could use look up similar structures. Oh wate, there is . Anyway, adamantane is not as regular as cubane, but it is highly symmetrical, looking like three cyclohexanes fused together. The vertices fall into two different types when colored by signature: The carbons with three carbon neighbours (degree-3, in the simple graph) have signature (a) and the degree-2 carbons have signature (b). Atoms of one type are only connected to atoms of another - the graph is bipartite . Adamantane connects together to form diamondoids (or, rather, this class have adamantane as a repeating subunit). One such is diamantane , which is no longer bipartite when colored by signature: It has three classes of vertex in the simple graph (a and b), as the set with degree-3 has been split in two. The tree for signature (c) is not shown. The graph is still bipartite accordin...

chalky

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1,2-dichlorocyclopropane and a spiran

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