### Canonical Augmentation : How not to do it

In the interest of not publishing false information (even on a blog), this is a brief outline of why the approach outlined in the previous post does not work.

When constructing a set of graphs - from their degree sequences or otherwise - it is usual to describe the process as traversing a tree. The root of the tree is the empty set, and the leaves are the completed graphs. Internal nodes in this tree are partially completed graphs, and parent nodes are connected to children by augmentations.

An augmentation to a graph can be made in various ways: adding a single edge, adding multiple edges to a single new node, etc. However, the important thing is to avoid duplicating solutions at the leaves.

One easy way to do this is to check all newly generated leaves against all the previously generated ones. However, this requires not only storage of all the solutions so far, but also a lot of isomorphism checks. A better way is to check if a solution is canonical by some means. Even better than that is to test the partial graphs (the internal tree nodes) and only follow paths of canonical augmentations.

So this image (click for bigger) shows an example of this failing for the degree sequence [3, 3, 3, 3, 3, 3] (or 36). The first two steps (A, B) are fine : v{0} is connected to v{1,2,3} and then v{1} is connected to v{4,5}. Following the left path, we get to partial graph (C) where v{2} is now connected to v{4,5} - unfortunately the vertices are now no longer in 'degree order'. Compare to the path (B-E-F) on the right to the prism. There, each partial solution has vertices in degree order.

In short, then; canonical augmentation only works if the method to check for a canonical solution is 'aligned' with the augmentation method. When augmenting by saturating orbits, it does not seem possible to use partition refinement to check a graph as canonical, since refinement always starts by partitioning by degree.

### 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…