### Duals and Inner Duals of Planar Graphs

Graphs that can be embedded in the plane are called planar graphs - that is, they can be drawn without crossings. These drawings (or 'embeddings') have a dual which also corresponds to a graph, with a vertex in the dual for every face in the original, and an edge in the dual for every pair of faces that share an edge.
For example, for each of the embeddings above (in black) of the same graph there is a different dual drawing corresponding to different graphs. If the vertex that represents the outer face is deleted from a dual, you get an inner dual.

These inner duals are what Brinkmann et al use for generating fusanes, as I mentioned. The basic idea is to generate the duals, and then expand those into fusanes. Since the (inner) duals necessarily have less vertices than the full graph, this is quite efficient.

However, I admit that I couldn't be bothered to implement the algorithm properly, even though the technique is partly based on McKay's method which I know a little about. Instead, I simply generated trees for the duals. These then have to be 'labeled' - confusingly this is not the same kind of labeling as for a labeled tree. Instead, the labels are the number of edges deleted from the dual to make the inner dual.

This image shows a single tree with two possible labelings, and the fusanes that correspond to those labelings. The middle vertices of the tree are labeled with a pair (1/3, 3/1, or 2/2) because there are two distinct parts to the outside curve. It might be obvious that the graph on the left - which corresponds to a labeling of 5,1/3,3/1,5 - would be identical to the one made by a labeling of 5,3/1,1/3,5. Or, to put it another way, you can rotate it through 180 degrees and get the same graph.

Here are all the fusanes made by this technique from tree-inner-duals on 7 vertices:

the far-left in the center is not a cycle, it's just the branches meet. There's also a duplicate in there!

### 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) …