Structural characterization of d-flattenability forbidden minors

Characterize, for each integer d ≥ 0, the family of d-flattenability forbidden minors by intrinsic structural properties (beyond merely listing them as the excluded-minor set for d-flattenable graphs).

Background

Flattenability in dimension d is a minor-closed property, so d-flattenable graphs admit a finite set of forbidden minors. While specific sets are known for small d (e.g., K5 and K2,2,2 for d = 3), a conceptual structural characterization of the obstructions themselves is lacking.

Such a characterization would likely be essential for extending the SIP characterization to higher dimensions without enumerating or relying on properties of large explicit forbidden-minor families.