User blog comment:Hyp cos/SCG(n) and some related/@comment-91.149.140.37-20160226224207/@comment-5529393-20160227024050

K(3,3) was the first nonplanar graph that Hyp Cos mentioned, so the answer to your first question would be, "Every nonplanar graph other than K(3,3)".

For your second question, Hyp Cos has yet to order all the planar graphs, so it's too soon to ask where the Petersen graph would fall in the ordering.