### Orbit Saturation of Degree Sequences

Degree sequences and signatures are related, so it seems reasonable that Faulon's orbit-saturation algorithm for enumerating molecules should also work for just degree sequences. For example, height-1 signatures for a carbon skeleton are practically identical to a degree sequence. See this image for details:

On the left is a target signature, with its corresponding carbon skeleton below. On the right is the equivalent degree sequence, and its simple graph. Of course, signatures can handle multiple bonds, and different elements, but the principle is the same.

So the major problem with Király's method for enumerating graphs from their degree sequences is that it produces isomorphic solutions. The problem I had with Faulon's method was that I couldn't get it to work! A functional implementation for the (much simpler) degree sequence problem should help.

Encouragingly, the initial crude implementation does work for simple examples. The key seems to be to  check that a solution is canonical by permuting within the original orbits, and not the final ones. The example I have been using is the sequence [3, 3, 2, 2, 1, 1] and one of its solutions is instructive here:

These two graphs are isomorphic, but the one on the left is the canonical form according to its adjacency matrix (below each graph). The matrix can be converted to a string by reading left-to-right and top-to-bottom (below each matrix). These strings can be compared to find a maximum one - which is the left hand one in this case as 11100... is greater than 10011... (etc).

To efficiently generate graphs, it is necessary to be able to test these strings without storing all the already generated isomorphs. A simple way to test for a canonical graph is to try all permutations of labels on the vertices; in other words, renumber the graph. Now clearly the only way to renumber the left hand graph here to get the right is to permute labels on vertices 0 and 1.

However, in the final graphs, these are in different orbits! So it may be that it is necessary to use the original orbits - which is just that induced by the degree sequence, or [{0, 1}, {2, 3}, {4, 5}]. Time will tell...

One final problem is that this simple canonical checking procedure is very expensive for large regular graphs. For [3, 3, 3, 3, 3, 3, 3, 3] (38) we get various graphs, including a cube and a cuneane-like graph, but the canonical check has to try 8 factorial (8!) permutations for each successful solution.

Luckily there is a solution - the good old partition refiner :) It seems to work so far.

### 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…

### Generating Dungeons With BSP Trees or Sliceable Rectangles

So, I admit that the original reason for looking at sliceable rectangles was because of this gaming stackoverflow question about generating dungeon maps. The approach described there uses something called a binary split partition tree (BSP Tree) that's usually used in the context of 3D - notably in the rendering engine of the game Doom. Here is a BSP tree, as an example:

In the image, we have a sliced rectangle on the left, with the final rectangles labelled with letters (A-E) and the slices with numbers (1-4). The corresponding tree is on the right, with the slices as internal nodes labelled with 'h' for horizontal and 'v' for vertical. Naturally, only the leaves correspond to rectangles, and each internal node has two children - it's a binary tree.

So what is the connection between such trees and the sliceable dual graphs? Well, the rectangles are related in exactly the expected way:

Here, the same BSP tree is on the left (without some labels), and the slicea…

### Listing Degree Restricted Trees

Although stack overflow is generally just an endless source of questions on the lines of "HALP plz give CODES!? ... NOT homeWORK!! - don't close :(" occasionally you get more interesting ones. For example this one that asks about degree-restricted trees. Also there's some stuff about vertex labelling, but I think I've slightly missed something there.

In any case, lets look at the simpler problem : listing non-isomorphic trees with max degree 3. It's a nice small example of a general approach that I've been thinking about. The idea is to:
Given N vertices, partition 2(N - 1) into N parts of at most 3 -> D = {d0, d1, ... }For each d_i in D, connect the degrees in all possible ways that make trees.Filter out duplicates within each set generated by some d_i. Hmm. Sure would be nice to have maths formatting on blogger....

Anyway, look at this example for partitioning 12 into 7 parts:

At the top are the partitions, in the middle the trees (colored by degree) …