Forbidden minors for 3-connected graphs with no non-splitting 5-configurations

For a set of five edges, a graph splits if one of the associated Dodgson polynomials is equal to zero. A graph G splitting for every set of five edges is a minor-closed property. As such there is a finite set of forbidden minors F such that if a graph H does not contain a minor isomorphic to any graph in F}, then H splits. In this paper we prove that if a graph G is simple, 3-connected, and splits, then G must not contain any minors isomorphic to K5, K3,3, the octahedron, the cube, or a graph that is a single Delta-Y transformation away from the cube. As such this is the set of all simple 3-connected forbidden minors. The complete set of 2-connected or non-simple forbidden minors remains unresolved, though a number have been found.