### Open Molecule Generator's Algorithm

With the recent release of the Open Molecule Generator (OMG) I thought it would be nice to add to (or augment) the description of the algorithm in the paper. The description here will be in terms of graphs, but the principle is largely the same.

OMG's algorithm is a variant of McKay's canonical path augmentation algorithm, as mentioned before here. However, instead of augmenting by vertex it augments by edges. To illustrate this, consider a diagram for graphs with up to 4 vertices:
The graphs are grouped by vertex count (boxes), and each box is sorted into columns. Along the top are the number of edges for graphs in that column, and along the bottom are number of vertices for graphs in the box. Two graphs are connected by an arrow if the larger can be made from the smaller by adding a single edge.

A couple of important things to note about edge-addition are : 1) two paths can lead to the same graph, and 2) at least one of the graphs is disconnected. The first of these is solved by canonical augmentation; a bond is only added if it is the inverse of a canonical deletion. For example:

The image above shows two parent graphs on the left, and an extension of each on their right. The extensions are isomorphic, and have the same canonical form on the far right. The canonical deletion edge is 2:3 - which maps to 0:3 in both graphs. Only one extension is equivalent to the canonical deletion edge.

The second problem - that of disconnected graphs - is only a problem if the canonicalisation routine cannot handle such graphs. Unfortunately, this currently applies to signatures; luckily OMG uses nauty for this part, which doesn't suffer from this limitation. An example of extending a disconnected graph is shown below:

Clearly connected graphs can be constructed from disconnected ones, although it is possible that a canonicalisation method could be designed that avoided disconnected parents. In other words, one that only chose edges that didn't separate the graph.

One final important point is that the OMG algorithm cannot proceed by extending every member of the set of graphs G(n) on n vertices to get non-redundant G(n + 1). For example, extending from both of the members of G(3) will produce multiple copies of one of G(4) - at least, in my test code. This might rule out one easy way to run the program in parallel (running separate subtrees in different processes), but possibly could be solved by using a subset of G(n) to generate the next level.

### 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) …

### Common Vertex Matrices of Graphs

There is an interesting set of papers out this year by Milan Randic et al (sorry about the accents - blogger seems to have a problem with accented 'c'...). I've looked at his work before here.

[1] Common vertex matrix: A novel characterization of molecular graphs by counting
[2] On the centrality of vertices of molecular graphs

and one still in publication to do with fullerenes. The central idea here (ho ho) is a graph descriptor a bit like path lengths called 'centrality'. Briefly, it is the count of neighbourhood intersections between pairs of vertices. Roughly this is illustrated here:

For the selected pair of vertices, the common vertices are those at the same distance from each - one at a distance of two and one at a distance of three. The matrix element for this pair will be the sum - 2 - and this is repeated for all pairs in the graph. Naturally, this is symmetric:

At the right of the matrix is the row sum (∑) which can be ordered to provide a graph invarian…