General d-dimensional SIP characterization via f-preserving forbidden minors

Prove that for any integer d ≥ 0, a graph–nonedge pair (G,f) has the d-single interval property (d-SIP) if and only if no atom of G ∪ f that contains f has an f-preserving d-flattenability forbidden minor.

Background

The paper establishes the equivalence for d ≤ 3 (with new results for d = 3), and suggests that the same structural criterion should hold in higher dimensions as well. Here, the d-flattenability forbidden minors denote the excluded-minor set characterizing d-flattenable graphs under Euclidean distance.

Extending to d ≥ 4 is challenging because the number and complexity of forbidden minors increase substantially (e.g., at least 75 for partial 4-trees are known), and a proof must avoid reliance on exhaustive properties of explicit obstructions.