General bounds for fault-tolerant variants of metric dimension

Determine what general relationships hold between the fault-tolerant parameter ftxdim(G) and the underlying parameter xdim(G) for arbitrary variants xdim(G) of metric dimension, including corresponding relationships for specified graph families and variants with additional structural properties.

Background

The paper proves the universal lower bound ftxdim(G) ≥ xdim(G)+1 and develops sharp or nearly sharp extremal results for several particular metric-dimension variants. It leaves open the broader question of identifying general principles governing fault tolerance across arbitrary variants and restricted graph classes such as trees, bipartite graphs, regular graphs, and degenerate graphs.

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