### 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.

### 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…

### Listing Degree Restricted Trees

Although stack overflow is generally just an endless source of questions on the lines of "HALP plz give CODES!? ... NOT homeWORK!! - don't close :(" occasionally you get more interesting ones. For example this one that asks about degree-restricted trees. Also there's some stuff about vertex labelling, but I think I've slightly missed something there.

In any case, lets look at the simpler problem : listing non-isomorphic trees with max degree 3. It's a nice small example of a general approach that I've been thinking about. The idea is to:
Given N vertices, partition 2(N - 1) into N parts of at most 3 -> D = {d0, d1, ... }For each d_i in D, connect the degrees in all possible ways that make trees.Filter out duplicates within each set generated by some d_i. Hmm. Sure would be nice to have maths formatting on blogger....

Anyway, look at this example for partitioning 12 into 7 parts:

At the top are the partitions, in the middle the trees (colored by degree) …

### Common Vertex Matrices of Graphs

There is an interesting set of papers out this year by Milan Randic et al (sorry about the accents - blogger seems to have a problem with accented 'c'...). I've looked at his work before here.

[1] Common vertex matrix: A novel characterization of molecular graphs by counting
[2] On the centrality of vertices of molecular graphs

and one still in publication to do with fullerenes. The central idea here (ho ho) is a graph descriptor a bit like path lengths called 'centrality'. Briefly, it is the count of neighbourhood intersections between pairs of vertices. Roughly this is illustrated here:

For the selected pair of vertices, the common vertices are those at the same distance from each - one at a distance of two and one at a distance of three. The matrix element for this pair will be the sum - 2 - and this is repeated for all pairs in the graph. Naturally, this is symmetric:

At the right of the matrix is the row sum (∑) which can be ordered to provide a graph invarian…