Characterize equality in the BKLPS Szeged–Wiener bound

Determine all integers n ≥ 10 and all n-unexceptional 2-connected graphs G of order n for which the Szeged–Wiener gap satisfies η(G) = 2n.

Background

The paper proves the BKLPS Szeged–Wiener Gap Conjecture: every n-unexceptional 2-connected graph of order n ≥ 10 satisfies η(G) ≥ 2n. It then constructs, for every n ≥ 10, an n-unexceptional 2-connected graph attaining equality, showing that the bound is sharp.

The authors explicitly leave unresolved the task of finding all equality cases. The problem asks for a necessary and sufficient characterization, rather than merely constructing examples; the subsequent construction is stated to provide only a sufficient condition and is noted not to be necessary.

References

We invite the reader to determine a necessary and sufficient condition for when equality holds, and we phrase it as a problem here.

Find all integers $n \ge 10$ and $n$-unexceptional $2$-connected graphs of order $n$ such that \eta(G) = 2n.

— Improved Bounds on the Szeged-Wiener Gap and the BKLPS Conjecture  (2609.20025 - Zhang et al., 17 Sep 2026) in Section 5, “Further Results,” immediately before the displayed Problem environment