### 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 right is the starting degree sequence - [3, 2, 2, 2, 1] - and there are two graphs at the bottom that realise this sequence. The 'tree of trees' in between is the recursive search through sets of neighbours for vertices in the graph. So the top tree shows the possible choices for neighbours of the last (4th) vertex; the next level shows them for the 3rd vertex, and so on.

The key point here is that only red leaves of a particular tree are valid choices, and these pass through a path of red and black edges. A red edge in the tree represents an edge in the graph, while a black edge indicates no edge from this vertex. So the left hand graph is [{0} : 4, {0, 1} : 3, {0, 1} : 2], using the notation {V0, V1, ..., Vn} : Vm for a set of edges {V0:Vm, V1:Vm, ..., Vn:Vm}. The final edge for the right hand graph (0:1) is not shown as a tree, since the degree sequence is [1, 1, 0, 0, 0] at that point - leaving only one choice.

Also not shown are colors on the internal nodes of the tree. Király's paper describes how to color these nodes so that the algorithm never visits any black leaf. This is vital for efficient output, but I have not ye implemented that part. However, cross-checking the results against the graphs output by McKay's method is promising so far (up to 7 vertices). I should note that Király's method seems to produce isomorphic solutions.

Code for this is here, although it is a somewhat naïve implementation.

Anonymous said…
very Good blogThank you!
Anonymous said…
Thank you for explaining the method. But seems the link for the code does not work.
gilleain said…
Sorry about that. Link is now fixed. Let me know if you have any problems with the code or suggestions to improve it.

### 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…