### McKay's canonical augmentation method explained for simple graphs

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...

Greg Kuperberg (mathoverflow.net/users/1450), Complete graph invariants?, http://mathoverflow.net/questions/11715 (version: 2010-01-14)
gilleain said…
I sort of understand what he's saying there. Actually, the code in the repository uses a certificate (which is a graph invariant) to do things like searching for a graph in a file or diff-ing two sets of graphs.
Anonymous said…
Hey there

### How many isomers of C4H11N are there?

One of the most popular queries that lands people at this blog is about the isomers of C4H11N - which I suspect may be some kind of organic chemistry question on student homework. In any case, this post will describe how to find all members of a small space like this by hand rather than using software.

Firstly, lets connect all the hydrogens to the heavy atoms (C and N, in this case). For example:

Now eleven hydrogens can be distributed among these five heavy atoms in various ways. In fact this is the problem of partitioning a number into a list of other numbers which I've talked about before. These partitions and (possible) fragment lists are shown here:

One thing to notice is that all partitions have to have 5 parts - even if one of those parts is 0. That's not strictly a partition anymore, but never mind. The other important point is that some of the partitions lead to multiple fragment lists - [3, 3, 2, 2, 1] could have a CH+NH2 or an NH+CH2.

The final step is to connect u…

### The Gale-Ryser Theorem

This is a small aside. While reading a paper by Grüner, Laue, and Meringer on generation by homomorphism they mentioned the Gale-Ryser (GR) theorem. As it turns out, this is a nice small theorem closely related to the better known Erdős-Gallai (EG).

So, GR says that given two partitions of an integer (p and q) there exists a (0, 1) matrixA iff p*dominatesq such that the row sum vector r(A) = p and the column sum vector c(A) = q.

As with most mathematics, that's quite terse and full of terminology like 'dominates' : but it's relatively simple. Here is an example:

The partitions p and q are at the top left, they both sum to 10. Next, p is transposed to get p* = [5, 4, 1] and this is compared to q at the bottom left. Since the sum at each point in the sequence is greater (or equal) for p* than q, the former dominates. One possible matrix is at the top left with the row sum vector to the right, and the column sum vector below.

Finally, the matrix can be interpreted as a bi…

### Havel-Hakimi Algorithm for Generating Graphs from Degree Sequences

A degree sequence is an ordered list of degrees for the vertices of a graph. For example, here are some graphs and their degree sequences:

Clearly, each graph has only one degree sequence, but the reverse is not true - one degree sequence can correspond to many graphs. Finally, an ordered sequence of numbers (d1 >= d2 >= ... >= dn > 0) may not be the degree sequence of a graph - in other words, it is not graphical.

The Havel-Hakimi (HH) theorem gives us a way to test a degree sequence to see if it is graphical or not. As a side-effect, a graph is produced that realises the sequence. Note that it only produces one graph, not all of them. It proceeds by attaching the first vertex of highest degree to the next set of high-degree vertices. If there are none left to attach to, it has either used up all the sequence to produce a graph, or the sequence was not graphical.

The image above shows the HH algorithm at work on the sequence [3, 3, 2, 2, 1, 1]. Unfortunately, this produce…