For the strip division I was using the stripper component, which automatically separates the mesh into strips one quad wide, in either the warp or weft direction (and if you want to switch between these, insert a MeshTurn component before the stripper).
stripperdemo.gh (14.0 KB)
However, a key property the mesh needs to have for this to be possible is all even valence vertices, which the outputs of the Fattener component do. Note that above the branching points are all surrounded by either 6 or 8 quads.
A mesh made from something like a union of boxes typically won’t have this property because there will be some valence 3 and 5 vertices at the corners, meaning there is no consistent warp/weft direction possible:
It’s a similar issue to what was shown there (though in that case, I think not having a consistent direction actually makes it more interesting)