Fault tolerance for metric dimension and its variants
Jesse Geneson, Shen-Fu Tsai
TL;DR
This work significantly sharpens our understanding of fault-tolerant metric dimension and its variants by establishing tighter exponential bounds and constructing families of graphs that tightly approach these bounds. The authors introduce a general strategy for building fault-tolerant resolving sets via closed neighborhoods and targeted augmentations, yielding near-optimal upper bounds for ftdim, ftedim, ftadim, and ftdim_k, with base-3 or base-2 growth depending on the variant. They provide sharp extremal results, including precise characterizations of graphs attaining minimum and maximum fault tolerance for several dimensions, and reveal deep connections between edge-metric extremality and Erdős–Kleitman-type problems in extremal set theory. A key highlight is the structural equivalence mc(k)=ek(k) between clique sizes in edge-mimension-bounded graphs and extremal-set families, linking graph theory to combinatorial set theory, and opening avenues for further cross-disciplinary insights.
Abstract
Hernando et al. (2008) introduced the fault-tolerant metric dimension $\text{ftdim}(G)$, which is the size of the smallest resolving set $S$ of a graph $G$ such that $S-\left\{s\right\}$ is also a resolving set of $G$ for every $s \in S$. They found an upper bound $\text{ftdim}(G) \le \dim(G) (1+2 \cdot 5^{\dim(G)-1})$, where $\dim(G)$ denotes the standard metric dimension of $G$. It was unknown whether there exists a family of graphs where $\text{ftdim}(G)$ grows exponentially in terms of $\dim(G)$, until recently when Knor et al. (2024) found a family with $\text{ftdim}(G) = \dim(G)+2^{\dim(G)-1}$ for any possible value of $\dim(G)$. We improve the upper bound on fault-tolerant metric dimension by showing that $\text{ftdim}(G) \le \dim(G)(1+3^{\dim(G)-1})$ for every connected graph $G$. Moreover, we find an infinite family of connected graphs $J_k$ such that $\dim(J_k) = k$ and $\text{ftdim}(J_k) \ge 3^{k-1}-k-1$ for each positive integer $k$. Together, our results show that \[\lim_{k \rightarrow \infty} \left( \max_{G: \text{ } \dim(G) = k} \frac{\log_3(\text{ftdim}(G))}{k} \right) = 1.\] In addition, we consider the fault-tolerant edge metric dimension $\text{ftedim}(G)$ and bound it with respect to the edge metric dimension $\text{edim}(G)$, showing that \[\lim_{k \rightarrow \infty} \left( \max_{G: \text{ } \text{edim}(G) = k} \frac{\log_2(\text{ftedim}(G))}{k} \right) = 1.\] We also obtain sharp extremal bounds on fault-tolerance for adjacency dimension and $k$-truncated metric dimension. Furthermore, we obtain sharp bounds for some other extremal problems about metric dimension and its variants. In particular, we prove an equivalence between an extremal problem about edge metric dimension and an open problem of Erdős and Kleitman (1974) in extremal set theory.