### Squashing Molecules Flat and Combinatorial Maps

So the traditional way to say it is probably 'planar embedding', but I've gone for the alternate terminology of 'squashing flat'. For many molecules, this is a fairly easy process : some decisions have to be made about rotatable bonds, but quite a few drugs are just things sticking off a benzene ring or two.

For fully 3-dimensional molecular graphs (for example) it is trickier. I don't yet know a simple algorithm for choosing different possible embeddings ... er ... squashings. Perhaps they are all complicated; they seem to involve tree data structures with names like "SPQR Tree" (see "Optimizing over all combinatorial embeddings of a planar graph").

Far easier, though, is the intermediate data structure between a graph and some 2D coordinates - known as a combinatorial map. These mathematical objects store the detail of the embedding by recording the order of 'half bonds' called flags (or darts?) around a vertex. Maybe a flag is the vertex and the half-bond. Anyway, here is a picture of K4 (or a squashed tetrahedrane skeleton):

Now the left hand side just shows the graph with labelled vertices, while the one on the right shows the flags (I'm going to call them that now, sorry). At the bottom is the permutation Ïƒ applied to the set of flags (f). To understand Ïƒ consider just the central vertex (1) : if we travel clockwise round from flag 1, we get the flags (1, 8, 6). In fact, this is how I constructed the permutation : clockwise round all the vertices. It can be helpful to consider Ïƒ in cycle notation as (0, 2, 4)(1, 6, 8)(5, 11, 9)(7, 10, 3) which has a cycle for each vertex.

This image shows two different embeddings of the same graph. Well, I suppose technically it is different embeddings of the same labelling. In any case, L and R are flipped : if there wasn't so much symmetry it might be better. The same procedure as before is used to get ÏƒL and ÏƒR (clockwise both times). Now check this out:

What is this crazy thing? Well, the other part of the combinatorial map is an 'involution' called Î±, which is really just another permutation that stores the pair of flags for each edge. In the images above, I have just ordered the edges using the vertices (as 0:1,0:2,0:3,1:2,1:3,2:3) then ordered the flags by vertex index : that is, Î± = [1, 0, 3, 2, 5, 4, 7, 6, 9, 8, 11, 10].

Anyway, using the formula Ï† = Ïƒ ⋅ Î± we get a permutation of what I actually am going to call 'darts'. This is all on the wikipedia page, but it should be clear that the labelled arrows on the image go in cycles. So there is (0, 6, 3) for example. Indeed Ï† = (0, 6, 3)(1, 4, 9)(2, 10, 5)(7, 8, 11) which is - of course - the four faces of the embedding, including the boundary face.

Phew! Fun stuff, and there is code here as well as some tests here. Well more like just System.out statements, but the beginning of tests...

### 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) …