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Square Grids, Cylinders, Spheres, and Toruses

This is straying from the point; but if any graph can be described by canonized trees made from its subgraphs, then what are the properties of very large (regular) graphs? A grid, for example?

Starting with a square, and fusing squares together results in this situation:


two and three fused squares are similar, at the height-two tree level. With four rings, a new type c appears (b becomes b' with three rings). Beyond this point, any number of rings fused in a row like this has the same structure.

Moving to a second dimension of growth, a square grid (Gnn) has the following structure:


On the left are 'snapshots' of the advancing wave of the tree. Looks a lot like a breadth-first search, I suppose. On the right are the trees for each snapshot, with increasing heights. Although this is not shown, 8 of the 20 leaves of the third tree are duplicates. This should be clear from the third snapshot, which has only 12 (filled) circles on the 'wavefront'.

These trees can even be extended to grids wrapped around three-dimensional objects. Or, to put it another way, grids wrapped up as surfaces:

The diamond shapes with lines radiating out are the expanding trees. To the left of each surface is a very rough sketch of what the spanning tree would be like. The dashed lines indicate cutpoints where the wavefronts meet - these are duplicated on the trees. This is similar to the idea of 'gluing' surfaces in topology.

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