### Canonical Augmentation : How not to do it

In the interest of not publishing false information (even on a blog), this is a brief outline of why the approach outlined in the previous post does not work.

When constructing a set of graphs - from their degree sequences or otherwise - it is usual to describe the process as traversing a tree. The root of the tree is the empty set, and the leaves are the completed graphs. Internal nodes in this tree are partially completed graphs, and parent nodes are connected to children by augmentations.

An augmentation to a graph can be made in various ways: adding a single edge, adding multiple edges to a single new node, etc. However, the important thing is to avoid duplicating solutions at the leaves.

One easy way to do this is to check all newly generated leaves against all the previously generated ones. However, this requires not only storage of all the solutions so far, but also a lot of isomorphism checks. A better way is to check if a solution is canonical by some means. Even better than that is to test the partial graphs (the internal tree nodes) and only follow paths of canonical augmentations.

So this image (click for bigger) shows an example of this failing for the degree sequence [3, 3, 3, 3, 3, 3] (or 36). The first two steps (A, B) are fine : v{0} is connected to v{1,2,3} and then v{1} is connected to v{4,5}. Following the left path, we get to partial graph (C) where v{2} is now connected to v{4,5} - unfortunately the vertices are now no longer in 'degree order'. Compare to the path (B-E-F) on the right to the prism. There, each partial solution has vertices in degree order.

In short, then; canonical augmentation only works if the method to check for a canonical solution is 'aligned' with the augmentation method. When augmenting by saturating orbits, it does not seem possible to use partition refinement to check a graph as canonical, since refinement always starts by partitioning by degree.

### 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…