General bounds relating fault tolerance and metric dimension variants

Determine general bounds for the fault-tolerant parameter ftxdim(G) in terms of xdim(G), both for arbitrary variants xdim(G) of metric dimension and for special graph families such as trees, bipartite graphs, regular graphs, and degenerate graphs.

Background

The paper proves the universal lower bound ftxdim(G) ≥ xdim(G) + 1 and develops sharp or near-sharp results for several specific variants. It leaves unresolved what additional general relationships hold for arbitrary metric-dimension variants and restricted graph classes.

References

For arbitrary variants xdim(G) of metric dimension, what else can be said in general about ftxdim(G) with respect to xdim(G)? The same question can be asked for special families of graphs G like trees, bipartite graphs, k-regular graphs, and k-degenerate graphs, as well as for variants xdim(G) of metric dimension with certain properties.

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