Complete the characterization of intermediate local metric dimensions

Complete the characterization of graphs with intermediate values of local metric dimension ldim(G) and fault-tolerant local metric dimension ftldim(G), beyond the cases ldim(G) ∈ {0,1} and the corresponding extremal cases ftldim(G) ∈ {1,2,n}.

Background

The paper gives characterizations for the minimum and maximum values of local metric dimension and fault-tolerant local metric dimension. The authors explicitly identify the remaining task of describing graphs attaining other values.

References

A remaining problem is to complete the characterization for other values of ldim(G) and ftldim(G).

— Fault tolerance for metric dimension and its variants  (2502.02731 - Geneson et al., 4 Feb 2025) in Section 6, paragraph following Corollary 6.5

Is there a local characterization of combinatorial dimension?

— Group actions on injective spaces and Helly graphs  (2307.00414 - Haettel, 2023) in Section 4, subsection “Geodesic bicombings”