### Colorful Expanding Triangulations and Sliceable Rectangular Graphs

There is a whole area of study on visualisations called cartograms - most appealing are the ones that make countries look like inflated or deflated balloons. The rectangular versions of these are less pretty, but more interesting to me from a graph theory perspective.

I came across this subject via an impressive masters thesis by Vincent Kusters : '
Characterizing Graphs with a Sliceable Rectangular Dual' … which is a title that will take some explaining. Firstly, what is a 'rectangular dual' when it's at home? Well check this out:

Clearly the thing on the left is a graph, and on the right is its rectangular dual - in fact, this is the smallest 'sliceable' dual. By sliceable, I mean that the white rectangles can be made by recursively slicing up a rectangle. For example, if a slice is like [{0, 3, 4, 5, 6}, {1, 2}] for making the first split into the areas of 1 and 2 on the right, and all the rest on the left. The next could be [{0}, {3, 4, 5, 6}] and so on.

The colors of the graph indicate a top/bottom cut in red, and a left/right cut in blue. So rectangles 0 and 1 share a left-right (blue) boundary, while 0 and 4 share a top-bottom (red) boundary. The square nodes [T, R, B, L] are the 'corners', and serve to anchor the dual. There's a lot of detail that I'm skipping here, but this is the broad picture.

Interestingly - for me - Kuster's work makes use of a program called Plantri made by none other than Gunnar Brinkmann and Brendan McKay. It generates planar triangulations - which rectangular duals are examples of - and then colors them to make proper duals. The way Plantri works is fairly familiar; canonical path augmentation but with a restricted set of operations to add vertices and edges:

Starting from K4 - the complete graph on 4 vertices - these 'expansions' are applied to graphs while rejecting duplicates using CPA. Now the thing that occurs to me is the possibility of expanding while maintaining the colorings of a rectangular dual. For example:

These are just examples of E5 from the picture before, but starting from particular colorings, and expanding only to particular colorings. As can be seen from the rectangular slices to the side of each graph, these expansions are 'compatible' in some sense with changes in the dual. Whether this is a meaningful operation or not, I'm not sure. There are a number of possible such expansions, but not a huge number. Here are a couple more:

Note that B and C are the same, but expand to different possible colorings. Also that the outer cycle colors are preserved, along with the some of the internal edges. That is no particular coincidence, since they were chosen specifically to preserve as many of the edges colors as possible.

Interesting, but not yet conclusive in any way.

### 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) …