Exact minimum order for fault cost 1

Determine the exact value of the minimum order of a 2-connected graph with fault cost 1, denoted by $\varphi_1$.

Background

The authors define φk\varphi_k as the order of the smallest graph with fault cost kk. Computational results establish the minimum orders for all listed values of k except 1, where the paper only narrows the value to 13 or 14. The appendix explicitly states that the exact value of φ1\varphi_1 is unknown.

References

A smallest graph attaining fault cost $k$ for $k\leq 8$, with the exception of $k = 1$ (we do not know the exact value of $\varphi_1$).

— Network fault costs based on minimum leaf spanning trees  (2502.10213 - Goedgebeur et al., 14 Feb 2025) in Caption of Figure "A smallest graph attaining fault cost $k$ for $k\leq 8$", Appendix, and Proposition 4.1 discussion