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Colorful Expanding Triangulations and Sliceable Rectangular Graphs

There is a whole area of study on visualisations called cartograms - most appealing are the ones that make countries look like inflated or deflated balloons. The rectangular versions of these are less pretty, but more interesting to me from a graph theory perspective.

I came across this subject via an impressive masters thesis by Vincent Kusters : '
Characterizing Graphs with a Sliceable Rectangular Dual' … which is a title that will take some explaining. Firstly, what is a 'rectangular dual' when it's at home? Well check this out:

Clearly the thing on the left is a graph, and on the right is its rectangular dual - in fact, this is the smallest 'sliceable' dual. By sliceable, I mean that the white rectangles can be made by recursively slicing up a rectangle. For example, if a slice is like [{0, 3, 4, 5, 6}, {1, 2}] for making the first split into the areas of 1 and 2 on the right, and all the rest on the left. The next could be [{0}, {3, 4, 5, 6}] and so on.

The colors of the graph indicate a top/bottom cut in red, and a left/right cut in blue. So rectangles 0 and 1 share a left-right (blue) boundary, while 0 and 4 share a top-bottom (red) boundary. The square nodes [T, R, B, L] are the 'corners', and serve to anchor the dual. There's a lot of detail that I'm skipping here, but this is the broad picture.

Interestingly - for me - Kuster's work makes use of a program called Plantri made by none other than Gunnar Brinkmann and Brendan McKay. It generates planar triangulations - which rectangular duals are examples of - and then colors them to make proper duals. The way Plantri works is fairly familiar; canonical path augmentation but with a restricted set of operations to add vertices and edges:


Starting from K4 - the complete graph on 4 vertices - these 'expansions' are applied to graphs while rejecting duplicates using CPA. Now the thing that occurs to me is the possibility of expanding while maintaining the colorings of a rectangular dual. For example:



These are just examples of E5 from the picture before, but starting from particular colorings, and expanding only to particular colorings. As can be seen from the rectangular slices to the side of each graph, these expansions are 'compatible' in some sense with changes in the dual. Whether this is a meaningful operation or not, I'm not sure. There are a number of possible such expansions, but not a huge number. Here are a couple more:


Note that B and C are the same, but expand to different possible colorings. Also that the outer cycle colors are preserved, along with the some of the internal edges. That is no particular coincidence, since they were chosen specifically to preserve as many of the edges colors as possible.

Interesting, but not yet conclusive in any way.

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