### McKay's canonical augmentation method explained for simple graphs

The previous post talked about generating one type of combinatorial object (chessboards) using a method similar to that outlined by Brendan McKay in a paper called "Isomorph-free exhaustive generation" (J Algorithms, 26 (1998) 306-324.). This one will focus instead on simple graphs, which requires both parts of the method.

The canonical construction (or canonical augmentation) method has two components. Firstly, only one 'expansion' of a graph is tried at each step from the set of equivalent expansions. Secondly, the expansions are checked to see if they are the inverse of a 'canonical deletion' for that graph.

For an example of the first rule, consider this set of expansions of a 4-vertex graph on the left:

Each of the 5-vertex graphs on the right are shown with the newly added vertex and edges in red; the arrows are labelled by the added edge set - so {1:4, 3:4} means edges added from 1 to 4 and 3 to 4. The sets of vertices to add to - {{0}, {1}, {1,3}} - are representatives of the orbit of these vertices. For example, the orbit of {1} in G(4) is {1, 2} as these two vertices are equivalent in G(4) on the left.

This is now quite similar to the situation with chessboards : trying only minimal orbit representatives for extending an object. In McKay's paper, the process of generating child objects is split into 'upper' and 'lower' objects. An upper object is a pair where X is (say) a graph, and W is a set of vertices to connect to a new vertex. A lower object is a pair where v is a vertex to delete. This is illustrated here:

Click for bigger, as usual. There is a function shown between a lower object for X' and an upper object for X. This is the 'deletion' function, and its inverse is the important one : f-1, the function that adds a new vertex by connecting it to all the vertices in W.

This process will generate isomorphic graphs, so there has to be a way to reject children that are not canonical. This is where the second part comes in ... unfortunately it is harder to describe.

Roughly, we need to check that the newly added vertex is the one that should have been added if it was canonical. To verify this, the child graph is canonically labelled (eg : see this post, or possibly this one) and then the code checks if the added vertex (under the canonical labelling) is in the same orbit as the last one. Kind of.

The upshot is that this code now produces results very similar to nauty (geng) for graphs up to 8-12 vertices. For the larger numbers, I started to restrict the maximum degree, to shorten the runtime. It's definitely not as fast as nauty, but not too bad. I still have the lingering suspicion that I might start missing graphs for larger spaces, but it's not bad, not bad at all...

Check out this reply..

Greg Kuperberg (mathoverflow.net/users/1450), Complete graph invariants?, http://mathoverflow.net/questions/11715 (version: 2010-01-14)
gilleain said…
I sort of understand what he's saying there. Actually, the code in the repository uses a certificate (which is a graph invariant) to do things like searching for a graph in a file or diff-ing two sets of graphs.
Anonymous said…
Hey there

### 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…

### 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…

### Signatures with user-defined edge colors

A bug in the CDK implementation of my signature library turned out to be due to the fact that the bond colors were hard coded to just recognise the labels {"-", "=", "#" }. The relevant code section even had an XXX above it!

Poor show, but it's finally fixed now. So that means I can handle user-defined edge colors/labels - consider the complete graph (K5) below:

So the red/blue colors here are simply those of a chessboard imposed on top of the adjacency matrix - shown here on the right. You might expect there to be at least two vertex signature classes here : {0, 2, 4} and {1, 3} where the first class has vertices with two blue and two red edges, and the second has three blue and two red.

Indeed, here's what happens for K4 to K7:

Clearly even-numbered complete graphs have just one vertex class, while odd-numbered ones have two (at least?). There is a similar situation for complete bipartite graphs:

Although I haven't explored any more of these…