User blog comment:Wythagoras/Graphs/@comment-1605058-20140911185546/@comment-5529393-20140912051503

Ah, thanks.

co-P5 is a subgraph of co-(P2 U P3), so it needn't be considered. co-(P2 U P3) is indeed a minor of K3,3 - the point on the left (in the picture from the website) is the contraction of two points.

The fact that you can follow K4 U K1 with K3,3 suggests that it might not be a bad way to start after all.