Orbit Saturation of Degree Sequences

Degree sequences and signatures are related, so it seems reasonable that Faulon's orbit-saturation algorithm for enumerating molecules should also work for just degree sequences. For example, height-1 signatures for a carbon skeleton are practically identical to a degree sequence. See this image for details:

On the left is a target signature, with its corresponding carbon skeleton below. On the right is the equivalent degree sequence, and its simple graph. Of course, signatures can handle multiple bonds, and different elements, but the principle is the same.

So the major problem with Király's method for enumerating graphs from their degree sequences is that it produces isomorphic solutions. The problem I had with Faulon's method was that I couldn't get it to work! A functional implementation for the (much simpler) degree sequence problem should help.

Encouragingly, the initial crude implementation does work for simple examples. The key seems to be to check that a solution is canonical by permuting within the original orbits, and not the final ones. The example I have been using is the sequence [3, 3, 2, 2, 1, 1] and one of its solutions is instructive here:

These two graphs are isomorphic, but the one on the left is the canonical form according to its adjacency matrix (below each graph). The matrix can be converted to a string by reading left-to-right and top-to-bottom (below each matrix). These strings can be compared to find a maximum one - which is the left hand one in this case as 11100... is greater than 10011... (etc).

To efficiently generate graphs, it is necessary to be able to test these strings without storing all the already generated isomorphs. A simple way to test for a canonical graph is to try all permutations of labels on the vertices; in other words, renumber the graph. Now clearly the only way to renumber the left hand graph here to get the right is to permute labels on vertices 0 and 1.

However, in the final graphs, these are in different orbits! So it may be that it is necessary to use the original orbits - which is just that induced by the degree sequence, or [{0, 1}, {2, 3}, {4, 5}]. Time will tell...

One final problem is that this simple canonical checking procedure is very expensive for large regular graphs. For [3, 3, 3, 3, 3, 3, 3, 3] (3⁸) we get various graphs, including a cube and a cuneane-like graph, but the canonical check has to try 8 factorial (8!) permutations for each successful solution.

Luckily there is a solution - the good old partition refiner :) It seems to work so far.

Comments

Adamantane, Diamantane, Twistane

After cubane, the thought occurred to look at other regular hydrocarbons. If only there was some sort of classification of chemicals that I could use look up similar structures. Oh wate, there is . Anyway, adamantane is not as regular as cubane, but it is highly symmetrical, looking like three cyclohexanes fused together. The vertices fall into two different types when colored by signature: The carbons with three carbon neighbours (degree-3, in the simple graph) have signature (a) and the degree-2 carbons have signature (b). Atoms of one type are only connected to atoms of another - the graph is bipartite . Adamantane connects together to form diamondoids (or, rather, this class have adamantane as a repeating subunit). One such is diamantane , which is no longer bipartite when colored by signature: It has three classes of vertex in the simple graph (a and b), as the set with degree-3 has been split in two. The tree for signature (c) is not shown. The graph is still bipartite accordin

Király's Method for Generating All Graphs from a Degree Sequence

After posting about the Hakimi-Havel theorem, I received a nice email suggesting various relevant papers. One of these was by Zoltán Király called " Recognizing Graphic Degree Sequences and Generating All Realizations ". I have now implemented a sketch of the main idea of the paper, which seems to work reasonably well, so I thought I would describe it. See the paper for details, of course. One focus of Király's method is to generate graphs efficiently , by which I mean that it has polynomial delay. In turn, an algorithm with 'polynomial delay' takes a polynomial amount of time between outputs (and to produce the first output). So - roughly - it doesn't take 1s to produce the first graph, 10s for the second, 2s for the third, 300s for the fourth, and so on. Central to the method is the tree that is traversed during the search for graphs that satisfy the input degree sequence. It's a little tricky to draw, but looks something like this: At the top

General Graph Layout : Putting the Parts Together

An essential tool for graph generation is surely the ability to draw graphs. There are, of course, many methods for doing so along with many implementations of them. This post describes one more (or perhaps an existing method - I haven't checked). Firstly, lets divide a graph up into two parts; a) the blocks, also known as ' biconnected components ', and b) trees connecting those blocks. This is illustrated in the following set of examples on 6 vertices: Trees are circled in green, and blocks in red; the vertices in the overlap between two circles are articulation points. Since all trees are planar, a graph need only have planar blocks to be planar overall. The layout then just needs to do a tree layout on the tree bits and some other layout on the embedding of the blocks. One slight wrinkle is shown by the last example in the image above. There are three parts - two blocks and a tree - just like the one to its left, but sharing a single articulation point. I had

Some Stuff

Search This Blog