MMNA graphs on eight vertices or fewer

We prove that the only minor minimal non-­‐apex (MMNA) graphs on eight vertices or fewer are the Jorgensen J1 graph and five Petersen family graphs, P8, K4,4 – e, K3,3,1, P7, and K6. We determine the non-­‐planar (7,12) graphs of minimum degree at least two. We argue that there are five planar triangulations on seven vertices. We make use of the Petersen graphs of eight and seven vertices: P8, K4,4 – e, K3,3,1, and P7, and determine all graphs on eight vertices with one of these Petersen graphs as a minor. With these in hand we give a complete classification of MMNA graphs on seven or fewer vertices. For graphs on eight vertices, we illustrate our proof by contradiction argument with a detailed analysis of (8,17) and (8,18) graphs constructed by adding a vertex to a non-­‐planar (7, 12) graph.