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### Duals and Inner Duals of Planar Graphs

Graphs that can be embedded in the plane are called planar graphs - that is, they can be drawn without crossings. These drawings (or 'embeddings') have a dual which also corresponds to a graph, with a vertex in the dual for every face in the original, and an edge in the dual for every pair of faces that share an edge.
For example, for each of the embeddings above (in black) of the same graph there is a different dual drawing corresponding to different graphs. If the vertex that represents the outer face is deleted from a dual, you get an inner dual.

These inner duals are what Brinkmann et al use for generating fusanes, as I mentioned. The basic idea is to generate the duals, and then expand those into fusanes. Since the (inner) duals necessarily have less vertices than the full graph, this is quite efficient.

However, I admit that I couldn't be bothered to implement the algorithm properly, even though the technique is partly based on McKay's method which I know a little about. Instead, I simply generated trees for the duals. These then have to be 'labeled' - confusingly this is not the same kind of labeling as for a labeled tree. Instead, the labels are the number of edges deleted from the dual to make the inner dual.

This image shows a single tree with two possible labelings, and the fusanes that correspond to those labelings. The middle vertices of the tree are labeled with a pair (1/3, 3/1, or 2/2) because there are two distinct parts to the outside curve. It might be obvious that the graph on the left - which corresponds to a labeling of 5,1/3,3/1,5 - would be identical to the one made by a labeling of 5,3/1,1/3,5. Or, to put it another way, you can rotate it through 180 degrees and get the same graph.

Here are all the fusanes made by this technique from tree-inner-duals on 7 vertices:

the far-left in the center is not a cycle, it's just the branches meet. There's also a duplicate in there!

### How many isomers of C4H11N are there?

One of the most popular queries that lands people at this blog is about the isomers of C4H11N - which I suspect may be some kind of organic chemistry question on student homework. In any case, this post will describe how to find all members of a small space like this by hand rather than using software.

Firstly, lets connect all the hydrogens to the heavy atoms (C and N, in this case). For example:

Now eleven hydrogens can be distributed among these five heavy atoms in various ways. In fact this is the problem of partitioning a number into a list of other numbers which I've talked about before. These partitions and (possible) fragment lists are shown here:

One thing to notice is that all partitions have to have 5 parts - even if one of those parts is 0. That's not strictly a partition anymore, but never mind. The other important point is that some of the partitions lead to multiple fragment lists - [3, 3, 2, 2, 1] could have a CH+NH2 or an NH+CH2.

The final step is to connect u…

### 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) matrixA iff p*dominatesq 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 the column sum vector below.

Finally, the matrix can be interpreted as a bi…

### Havel-Hakimi Algorithm for Generating Graphs from Degree Sequences

A degree sequence is an ordered list of degrees for the vertices of a graph. For example, here are some graphs and their degree sequences:

Clearly, each graph has only one degree sequence, but the reverse is not true - one degree sequence can correspond to many graphs. Finally, an ordered sequence of numbers (d1 >= d2 >= ... >= dn > 0) may not be the degree sequence of a graph - in other words, it is not graphical.

The Havel-Hakimi (HH) theorem gives us a way to test a degree sequence to see if it is graphical or not. As a side-effect, a graph is produced that realises the sequence. Note that it only produces one graph, not all of them. It proceeds by attaching the first vertex of highest degree to the next set of high-degree vertices. If there are none left to attach to, it has either used up all the sequence to produce a graph, or the sequence was not graphical.

The image above shows the HH algorithm at work on the sequence [3, 3, 2, 2, 1, 1]. Unfortunately, this produce…