Final post on Combinatorial Maps of Cuneane

EDIT : Updated with new image.
Well, unless I can make a more horrifying diagram than this:

..but I think that's unlikely. An explanation - if such a thing is possible - is that the red/purple arrows correspond to a kind of inside-out operation. So the triangular map (M3) at the bottom can be converted to the square (M4) by choosing the purple darts (2, 14, 16, 5) as the boundary, and putting everything else on the inside. The cycle (1, 4, 9, 20, 22) is red because it is the boundary of the pentagonal map (M5).

So, it seems like Ï•(M4) is Î±(Ï•(M5)) - so the cycle (0,8,23,21,5) in Ï•(M4) is (1,9,22,20,4) in Ï•(M5). That is, because (Î±[0] = 1, Î±[8] = 9, Î±[23] = 22 ...). However Ï•(M3) = Ï•(M5), which is confusing. Oh well

So I have updated the image so that all three 'maps' have the same cycles. Which means they have the same Ï• and therefore the same Ïƒ. I guess this means I misunderstood what a CM actually is : you can get 'different' embeddings with the same Ïƒ. Here is a summary diagram:

Which is just the same diagram, without all the numbers. Colored arrows in the faces show cycles of darts chosen as the bounding faces according to the same colored arrows joining maps. Note that the 4-cycle in CM3 chosen as the bounds for CM4 has to be flipped. So does the 5-cycle in CM3 chosen as the bounds of CM5. Perhaps there is still a transformation wrong here somewhere.

ANOTHER EDIT : After flipping CM5, and labelling the cycles (A-F/a-f : clockwise is uppercase, anticlockwise is lowercase) it is better.

The transitions are quite simple, really. Take a clockwise face, and make it anticlockwise (eg: B->b, red arrow from CM3 to CM5) and make the anticlockwise outer face clockwise (d -> D). These changes are on the red/purple arrows between maps. Under each map is a summary of the cycles, which really just shows that the outer face is anticlockwise and the others are clockwise.

chembioinfo said…
This provokes a lot of ideas and question in ones mind.
It also reminds me of flow optimisation problem in computer science.
Anonymous said…
I love this game :)

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…