Optimal density constant for metric graphs

Determine whether the constant (1/12)·e(G) in Theorem \ref{thm:optimal-graph} is optimal for compact, connected metric graphs G. Specifically, under the hypothesis that the subset X ⊂ G satisfies d_H(G,X) > \vec{d}_H(∂G,X) when ∂G ≠ ∅, prove or refute the conjecture that the lower bound can be strengthened to d_GH(G,X) ≥ min{ d_H(G,X), (1/8)·e(G) }, i.e., show that if d_GH(G,X) < (1/8)·e(G) then d_GH(G,X) ≥ d_H(G,X).

Background

Theorem \ref{thm:optimal-graph} establishes that for a compact, connected metric graph G with shortest non-terminal edge length e(G), and a subset X ⊂ G satisfying a boundary-density condition (d_H(G,X) > \vec{d}_H(∂G,X) when ∂G ≠ ∅), one has d_GH(G,X) ≥ min{ d_H(G,X), (1/12)·e(G) }. This yields equality d_GH(G,X) = d_H(G,X) whenever d_GH(G,X) < (1/12)·e(G).

The authors ask whether the constant (1/12)·e(G) is optimal and conjecture an improvement to (1/8)·e(G), which would strengthen the lower bound and the equality threshold for sufficiently dense subsets of metric graphs. This problem concerns sharpening the quantitative relationship between Gromov–Hausdorff and Hausdorff distances in the graph setting.