Generating Chessboards With K-Independant Vertex Sets

After looking at Julio Peironcely's poster on generating chemical structures, where he describes using Canonical Path Augmentation (I think due to Brendan McKay) I went looking for more about it. One thing I found was this talk/slideshow by Mathieu Dutour Sikirić - incidentally coauthor of a nice book on Chemical Graphs.

Anyway; that's the context. Now : about chessboards? Well one of the examples given for augmentation (or 'orderly') schemes of independent vertex sets. I'm not sure what made me think of chessboards for this, but I think it's a fairly standard simple toy example. Let me show an example for 3x3 boards:

So, these are all the three by three boards where no two black squares share an edge. In other words, if the board is considered as a grid-shaped graph, then these are the k-independent vertex sets. So the question is : how to generate these?

The simple way, of course, is just to fill in every square and eliminate those boards that have pairs of black squares across an edge. This is the 'brute force' approach, and scales badly : there are 2⁹ boards but we only want 20 of these, or just 3%. For 4x4 boards, this fraction is even smaller - 131 of 2¹⁶ boards which is only 0.19%. This is 0.0026% for 5x5 boards.

So, the number of boards increases rapidly as the size increases and any way of decreasing this large search space is essential. The approach outlined in Mathieu's talk is quite simple : only try sets (boards) that are the minimal representative in their orbit. Ok, so maybe that doesn't sound so simple :) but look at this:

Here are 3x3 boards, with a numbering (any one will do), the equivalence classes of the cells, and a set of orbits. Lets say we've just generated {0, 4, 6} : how do we check if it is the minimal representative? Well, so long as we have the automorphism group of the board we just apply each permutation in the group to the set of numbers and check to see if any are smaller. In this case, {0, 2, 4} is smaller, so we don't consider {0, 4, 6}.

It's really as simple as that. Of course, we have to check each newly added cell to make sure it is not adjacent, but that can be done without consideration of previously generated boards. This means we don't have to store solutions, which means it can be done in parallel without communication. See how many there are just for 5x5 boards:

I suppose that one last question about all this is : so what? Does this have anything to do with chemistry? Well, actually, it does. Consider the bond order assignment problem : the second image shows all the possible assignments, which you may notice has duplicates (3, 5, 10, and 12 for example). Also consider the line graph approach to double bond systems where again the second image shows a pair of colorings of line graphs. In fact, these are k-independent sets...

Oh, and there is code here