Compare fault tolerance with maximum degree across variants

Determine for which variants xdim(G) of metric dimension the exponential growth rate of the maximum fault-tolerant parameter ftxdim(G), as a function of xdim(G), equals the corresponding exponential growth rate of the maximum degree A(G).

Background

The paper observes that, for the metric-dimension variants analyzed, the growth of the largest possible fault-tolerant parameter is closely connected to the growth of the maximum degree. The authors ask whether this relationship persists for other variants and formulate the question as an equality between two asymptotic logarithmic growth rates.

References

Based on the observations about fault tolerance and maximum degree in this paper, we have the following question. For what variants xdim(G) of metric dimension is it true that lim k700 max G: xdim(G)=k log(ftxdim(G)) k # lim G: xdim(G)=k max log(A(G)) k )?

— Fault tolerance for metric dimension and its variants  (2502.02731 - Geneson et al., 4 Feb 2025) in Section 9, Conclusion

For what variants xdim(G) of metric dimension is it true that lim k700 max G: xdim(G)=k log(ftxdim(G)) k # lim G: xdim(G)=k max log(A(G)) k )?

— Fault tolerance for metric dimension and its variants  (2502.02731 - Geneson et al., 4 Feb 2025) in Conclusion, Section 9