Skip to main content

Graph Layout

Finally, after much effort, the planar embedder is working.

Along the way, a couple of other things have come out of this work : one is an implementation of a fusane generator roughly related to Brinkmann, Caporossi, and Hansen's paper. I say 'roughly' - actually, it is nothing like it! (To be described in a later post.)

I now also understand better algorithms for spanning tree generations, and cycle finding. These will also be described in later posts, if necessary. For now, though : what about those planar graphs, eh? Got any images...

Fusanes! Or, well, graphs that could model fusanes (polyhexes), to be exact. The colors here are vertex equivalence classes, calculated using signatures. These are quite easy examples, however - they are all outerplanar graphs. So what about something more tricky? How about this :

Not very nicely laid out, but looks a lot like the top picture here, I think. One lesson I have learnt is that an embedding (which is a combinatorial object) is not at all like a drawing (which is a geometric object). the embedding only tells you which vertices are part of which face, while the drawing has actual 2D coordinates.

There is an implementation of Plestenjak's spring layout algorithm - but I must have messed up somewhere, as it doesn't work so well. Take a look at this 'before and after' image of fullerene-26 :

It's a bit hard to see at this size, but the one on the left is the 'before' picture, where no refinement of the coordinates has been performed, and the one on the right has been refined. Well 'coarsened' in some places - notably the vertex at the top, which happens to be the central one.

Code for this is here.


Popular posts from this blog

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…

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…

Generating Dungeons With BSP Trees or Sliceable Rectangles

So, I admit that the original reason for looking at sliceable rectangles was because of this gaming stackoverflow question about generating dungeon maps. The approach described there uses something called a binary split partition tree (BSP Tree) that's usually used in the context of 3D - notably in the rendering engine of the game Doom. Here is a BSP tree, as an example:

In the image, we have a sliced rectangle on the left, with the final rectangles labelled with letters (A-E) and the slices with numbers (1-4). The corresponding tree is on the right, with the slices as internal nodes labelled with 'h' for horizontal and 'v' for vertical. Naturally, only the leaves correspond to rectangles, and each internal node has two children - it's a binary tree.

So what is the connection between such trees and the sliceable dual graphs? Well, the rectangles are related in exactly the expected way:

Here, the same BSP tree is on the left (without some labels), and the slicea…