### Fullerene Layout with Spokes and Arches

Having tried (and failed) to layout fullerene structures using various optimisation methods, I thought I would try direct positioning of the atoms. In other words, 'logical' placement rather than 'physics' based layout. For example:

These are two regular fullerenes that work very well. The algorithm is simple in principle:

1) Given a planar embedding G, calculate the inner dual id(G) and the 'face layers'.
2) The innermost layer is the 'core' which is one of: a single vertex, a connected pair, or a cycle.
3) Layout the core, and then each layer outwards, by spoke and arch.

So, to explain some of this; a 'face layer' is a set of faces all at the same distance from the outer cycle, measured by graph distance on id(G). So the faces adjacent to the outer cycle are the first layer, and the second layer is adjacent to that, and so on. This is roughly illustrated here:

The concentric circles represent the layers of faces, with the innermost being the core. On the right is a cartoon of two spokes on an outer path, connected by a dashed line to show the arch. The outer path starts off as the edges around the core, and is replaced by the arches of the next level.

This method does work - it creates diagrams that are not too horrible - but does not give as good results for the less-symmetric examples. Something like Bojan Mohar's circle-packing method would probably be better, or even a properly implemented spring layout...

Here are some final examples, all with paired-cores:

(The colors are signature classes, by the way. All images made using CDK's new classes. Code here).

### 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) …