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Showing posts from August, 2011

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 …

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 blac…