Sharpen the exponential fault-tolerance bounds

Sharpen the upper and lower bounds for the maximum fault-tolerant metric dimension, edge metric dimension, and truncated metric dimension relative to their corresponding dimensions, improving the currently known estimates beyond agreement only in the exponential base.

Background

The paper establishes exponential upper and lower bounds for fault-tolerant variants of metric dimension, edge metric dimension, and truncated metric dimension. Although the bounds determine the exponential base in several cases, the authors state that the estimates remain non-sharp in their finer terms and leave their further sharpening unresolved.