The forbidden-minor characterization conjecture for realizable dimension at most two

Let $G$ be a multigraph, let $\operatorname{rd}(G)$ be its realizable dimension, and let $\lambda(G)$ be the graph parameter defined in the paper. Let $\mathcal{F}$ be the finite collection of graphs displayed in Figure~.

Forbidden-minor characterization conjecture. The following are equivalent:

1. $\operatorname{rd}(G)\leq 2$.
2. $\lambda(G)\leq 2$.
3. $G$ has no minor isomorphic to a graph in $\mathcal{F}$.

Characterizing multigraphs with realizable dimension at most two is identified as an important open problem. The conjecture proposes both an equality with the parameter $\lambda$ at this threshold and a finite forbidden-minor description.