The edge-deletion characterization of forbidden flattenability minors

Let $d \geq 2$. A graph is a forbidden minor for $d$-flattenability when it is a minimal obstruction to $d$-flattenability. For an edge $e$ of a graph $G$, $(G \setminus e,e)$ denotes the graph-nonedge pair obtained by deleting $e$. Forbidden-minor SIP characterization. For any dimension $d \geq 2$, a graph $G$ is a forbidden minor for $d$-flattenability if and only if both of the following hold: for every edge $e$ of $G$, $(G \setminus e,e)$ does not have the $d$-SIP; and for every minor $[G]$ obtained by a single edge deletion or contraction, and every edge $e$ of $[G]$, $([G] \setminus e,e)$ has the $d$-SIP. This is proposed as a way to avoid relying on an explicit list of higher-dimensional forbidden minors.