Forbidden minors for graphs with no first obstruction to parametric Feynman integration

We give a characterization of 3-connected graphs which are planar and forbid cube, octahedron, and HH minors, where HH is the graph which is one Δ−YΔ−Y away from each of the cube and the octahedron. Next we say a graph is Feynman 5-split if no choice of edge ordering gives an obstruction to parametric Feynman integration at the fifth step. The 3-connected Feynman 5-split graphs turn out to be precisely those characterized above. Finally we derive the full list of forbidden minors for Feynman 5-split graphs of any connectivity.