Two Different Ways to Handle Disconnected Graphs when Generating Graphs with Signatures

Disconnected graphs were mentioned in this post on OMG's algorithm. I also said that signatures could not handle such graphs, but this is not totally true. Here are a couple of approaches to handle them.

Firstly, why is this a problem at all? Well, a signature of a graph is a bit like a spanning tree - and the canonical signature is the maximal string form of the set of trees. However, for disconnected graphs you get a spanning forest - which is just a set of spanning trees. The maximum tree from the forest will no longer span the whole graph.

The image shows an example of this for a very simple graph. The upper panel shows a disconnected graph and its corresponding forest. The lower panel shows two different labelings of the same graph (A, B) and their canonical form. The algorithm for canonicalization simply has to 'paste' together canonical labels or signature strings, and has to make sure that a canonical signature is generated for each component. The components are ordered by their canonical strings.

An alternative solution to the problem for generating graphs is mentioned in a blog post by Derrick Stolee. Simply choose the canonical deleted edge to be one that will not disconnect the graph. For example, with an edge-addition canonical path algorithm, any edge that is a bridge (as discussed yesterday) would disconnect the graph if deleted. So the canonical edge is no longer the last edge in the canonically labelled graph but the last non-bridge edge.

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chalky

I wanted to show something that hints at the things that the new architecture can afford us: This is using a Java2D graphics Paint object to make it look like chalk...kindof. It's a very simplistic way of doing it by making a small image with a random number of white, gray, lightgray, and black pixels. edit: it doesn't look so good at small scales some tweaking of stroke widths and so on is essential.

The Gale-Ryser Theorem

This is a small aside. While reading a paper by Grüner, Laue, and Meringer on generation by homomorphism they mentioned the Gale-Ryser (GR) theorem. As it turns out, this is a nice small theorem closely related to the better known Erdős-Gallai (EG). So, GR says that given two partitions of an integer ( p and q) there exists a (0, 1) matrix A iff p* dominates q such that the row sum vector r(A) = p and the column sum vector c(A) = q . As with most mathematics, that's quite terse and full of terminology like 'dominates' : but it's relatively simple. Here is an example: The partitions p and q are at the top left, they both sum to 10. Next, p is transposed to get p* = [5, 4, 1] and this is compared to q at the bottom left. Since the sum at each point in the sequence is greater (or equal) for p* than q , the former dominates. One possible matrix is at the top left with the row sum vector to the right, and th...

1,2-dichlorocyclopropane and a spiran

As I am reading a book called "Symmetry in Chemistry" (H. H. Jaffé and M. Orchin) I thought I would try out a couple of examples that they use. One is 1,2-dichlorocylopropane : which is, apparently, dissymmetric because it has a symmetry element (a C2 axis) but is optically active. Incidentally, wedges can look horrible in small structures - this is why: The box around the hydrogen is shaded in grey, to show the effect of overlap. A possible fix might be to shorten the wedge, but sadly this would require working out the bounds of the text when calculating the wedge, which has to be done at render time. Oh well. Another interesting example is this 'spiran', which I can't find on ChEBI or ChemSpider: Image again courtesy of JChempaint . I guess the problem marker (the red line) on the N suggests that it is not a real compound? In any case, some simple code to determine potential chiral centres (using signatures) finds 2 in the cyclopropane structure, and 4 in the ...

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