Existence and characterization of 3-connected fault-cost-one graphs

Determine whether 3-connected traceable graphs with fault cost 1 exist, characterize the case of connectivity 3 if such graphs exist, and determine whether k-leaf-guaranteed graphs with fault cost 1 exist for any integer k≥3.

Background

The paper develops structural constructions for graphs with fault cost 1, but all of its constructed traceable fault-cost-one (tfc1) graphs have connectivity 2 because they are based on 2-fragments. The authors therefore ask whether the phenomenon persists in the more restrictive setting of 3-connected graphs and, if so, whether the connectivity-3 case admits a structural characterization. They also pose the broader existence question for k-leaf-guaranteed graphs with fault cost 1 when k is at least 3.

References

Up to now we have been discussing constructions based on 2-fragments, thus all of our tfc1 graphs are of connectivity 2. It is natural to ask whether 3-connected tfc1 graphs exist. If they do, is there a characterization for (say) the connectivity 3 case? To move even a bit further we might also ask whether $k$-leaf-guaranteed graphs with fault cost 1 exist for $k\geq 3$.

— Network fault costs based on minimum leaf spanning trees  (2502.10213 - Goedgebeur et al., 14 Feb 2025) in Problem 1, Section 5 (Open problems)