Skip to main content

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.

Popular posts from this blog

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…

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…

Signatures with user-defined edge colors

A bug in the CDK implementation of my signature library turned out to be due to the fact that the bond colors were hard coded to just recognise the labels {"-", "=", "#" }. The relevant code section even had an XXX above it!

Poor show, but it's finally fixed now. So that means I can handle user-defined edge colors/labels - consider the complete graph (K5) below:

So the red/blue colors here are simply those of a chessboard imposed on top of the adjacency matrix - shown here on the right. You might expect there to be at least two vertex signature classes here : {0, 2, 4} and {1, 3} where the first class has vertices with two blue and two red edges, and the second has three blue and two red.

Indeed, here's what happens for K4 to K7:

Clearly even-numbered complete graphs have just one vertex class, while odd-numbered ones have two (at least?). There is a similar situation for complete bipartite graphs:

Although I haven't explored any more of these…