Forbidden-minor characterization of strictly metrizable graphs

Establish whether a finite graph is strictly metrizable if and only if it contains none of the six graphs depicted in Figure minor_zoo as a minor, namely the six currently proposed minor-minimal non-strictly-metrizable graphs.

Background

The paper proves that strictly metrizable graphs form a minor-closed family. By the Graph Minor Theorem, every nontrivial minor-closed graph family is characterized by a finite set of forbidden minors, but the authors do not determine that forbidden-minor set for strict metrizability.

The conjecture proposes that the six graphs displayed in Figure minor_zoo constitute exactly the forbidden minors. Resolving it would determine which subdivisions of the relevant graphs—particularly subdivisions of K_{2,3}, K_4, W_4, and W_4'—are strictly metrizable and could clarify the structure of metrizable graphs.

References

We raise the possibility that answer is to be found in \cref{fig:minor_zoo}. Namely, \begin{conjecture}\label{conj:gang_of_six} A graph is strictly metrizable if and only if it contains none of the six graphs in \cref{fig:minor_zoo} as a minor. \end{conjecture}

— Strictly Metrizable Graphs are Minor-Closed  (2501.08277 - Chudnovsky et al., 14 Jan 2025) in Conjecture 1, Section Discussion and Open Questions