Characterize variants whose fault-tolerance and maximum-degree growth coincide

Determine for which variants xdim(G) of metric dimension the exponential growth rate of the maximum fault-tolerant parameter ftxdim(G), measured over graphs with xdim(G) = k, equals the corresponding exponential growth rate of the maximum vertex degree A(G).

Background

For the metric-dimension variants analyzed in the paper, the authors observe that the maximum fault-tolerant value grows on essentially the same exponential scale as the corresponding maximum degree. They formulate a general limit comparison to identify precisely which other variants share this behavior.

The problem asks for a classification of metric-dimension variants satisfying the displayed equality of asymptotic logarithmic growth rates.