User blog comment:Wythagoras/Extension of SCG/@comment-5529393-20130808115936

You're forgetting again that a loop is a minor of a k-cycle. So the loop goes below the triangle (and the 2-cycle) and corresponds to f_omega (x).

For SCG(1,2), we can use:

G2: 2-cycle

G3: 3 loops

G4: 2 loops + stick

G5: 2 loops + 3 dots

G6: 2 loops + 2 dots

G7: 2 loops + 1 dot

G8: 2 loops

G9: 1 loop plus 8 path

G10: 1 loop, 7 path, stick

G11: 1 loop, 7 path, 3 dots

G14: 1 loop, 7 path

G15: 1 loop, 2 6-paths, stick

G16: 1 loop, 2 6-paths, 3 dots

G19: 1 loop, 2 6-paths

Here I mean an n-path to have n vertices, n-1 edges.

Alternatively, we can go

G2: 2 loops

G3: triangle

G4: 2-cycle + stick

G5: 2-cycle + 3 dots

G8: 2-cycle

G9: 1 loops plus 8-path

and the rest is the same.

So SCG(1,2) ~ f_omega (f_5 (f_5 (3))