Sharpen the exponential bounds for fault-tolerant metric dimensions

Sharpen the upper and lower bounds for the maximum fault-tolerant metric dimension, fault-tolerant edge metric dimension, and fault-tolerant truncated metric dimensions so that the exponential bases are determined exactly.

Background

The paper establishes exponential upper and lower bounds for several fault-tolerant variants of metric dimension. Although the bounds have matching growth orders, the authors note that the bases of the exponent are not fully matched in all cases, leaving open the problem of determining sharper extremal estimates.