The previous post talked about generating one type of combinatorial object (chessboards) using a method similar to that outlined by Brendan McKay in a paper called "*Isomorph-free exhaustive generation*" (J Algorithms, 26 (1998) 306-324.). This one will focus instead on simple graphs, which requires both parts of the method.

The *canonical construction* (or *canonical augmentation*) method has two components. Firstly, only one 'expansion' of a graph is tried at each step from the set of equivalent expansions. Secondly, the expansions are checked to see if they are the inverse of a 'canonical deletion' for that graph.

For an example of the first rule, consider this set of expansions of a 4-vertex graph on the left:

Each of the 5-vertex graphs on the right are shown with the newly added vertex and edges in red; the arrows are labelled by the added edge set - so {1:4, 3:4} means edges added from 1 to 4 and 3 to 4. The sets of vertices to add to - {{0}, {1}, {1,3}} - are representatives of the orbit of these vertices. For example, the orbit of {1} in G(4) is {1, 2} as these two vertices are equivalent in G(4) on the left.This is now quite similar to the situation with chessboards : trying only minimal orbit representatives for extending an object. In McKay's paper, the process of generating child objects is split into 'upper' and 'lower' objects. An upper object is a pair where X is (say) a graph, and W is a set of vertices to connect to a new vertex. A lower object is a pair where v is a vertex to delete. This is illustrated here:

Click for bigger, as usual. There is a function shown between a lower object for X' and an upper object for X. This is the 'deletion' function, and its inverse is the important one : f

^{-1}, the function that adds a new vertex by connecting it to all the vertices in W.

This process will generate isomorphic graphs, so there has to be a way to reject children that are not canonical. This is where the second part comes in ... unfortunately it is harder to describe.

Roughly, we need to check that the newly added vertex is the one that should have been added if it was canonical. To verify this, the child graph is canonically labelled (eg : see this post, or possibly this one) and then the code checks if the added vertex (under the canonical labelling) is in the same orbit as the last one. Kind of.

The upshot is that this code now produces results very similar to nauty (geng) for graphs up to 8-12 vertices. For the larger numbers, I started to restrict the maximum degree, to shorten the runtime. It's definitely not as fast as nauty, but not too bad. I still have the lingering suspicion that I might start missing graphs for larger spaces, but it's not bad, not bad at all...

## Comments

Greg Kuperberg (mathoverflow.net/users/1450), Complete graph invariants?, http://mathoverflow.net/questions/11715 (version: 2010-01-14)