Fault tolerance for metric dimension and its variants

Jesse Geneson; Shen-Fu Tsai

REVIEW 2 major objections 5 minor 25 references

Fault tolerance for metric dimension and its variants

T0 review · 2 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read Fault-tolerant metric dimension grows exponentially with base 3, not 2 or 5, in the worst case.

desk verdict The paper settles the exponential base for fault-tolerant metric dimension (it's 3) and proves a clean equivalence with the Erdős–Kleitman problem; the gaps are presentational, not mathematical. read the letter →

arxiv 2502.02731 v1 pith:6BVT7D57 submitted 2025-02-04 math.CO cs.DM

Jesse Geneson , Shen-Fu Tsai This is my paper · ORCID

classification math.COcs.DM MSC 05C1205C35

keywords metricdimensionfault-tolerantedgeadjacencytruncatedresolvingsetErdős–Kleitmanproblemextremalgraphtheory

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Fault tolerance is expensive in metric dimension: if a graph needs $k$ landmarks to locate every vertex, it can need roughly $3^k$ landmarks before any one of them is allowed to fail. This paper proves that the worst-case exponential base is exactly 3, improving the previous upper bound of about $5^k$ and the previous lower bound of about $2^k$. The upper bound comes from enlarging any resolving set by its closed neighborhood, and the matching lower bound comes from a family of ternary-labeled stars $J_k$ in which every fault-tolerant resolving set must contain a vertex from almost every digit class. The same strategy yields the base-2 analogue for edge metric dimension and sharp bounds for adjacency, truncated, and local metric dimensions, and it connects the clique number of edge-metric-dimension-$k$ graphs to the Erdős–Kleitman problem on set families with distinct pairwise unions.

What carries the argument

The upper bound is carried by the closed-neighborhood enlargement of a resolving set: from any resolving set $S$, take $S$ together with all neighbors of vertices in $S$, then add one extra vertex for each landmark to repair the single unresolved pair that can remain; the bound that a landmark in a resolving set of size $k$ has at most $3^{k-1}$ neighbors controls the total size. The matching lower bound is carried by the family $J_k$: a star $K_{1,3^k}$ whose leaves are labeled by distinct ternary strings of length $k$, plus vertices $s_i,r_i$ that read the $i$-th ternary digit. In $J_k$, two leaves whose labels agree except in the first digit (0 versus 1) are distinguished only by themselves and by $s_1$, which forces any fault-tolerant resolving set to spend one vertex on almost every one of the $3^{k-1}$ digit classes.

What would settle it

Compute the exact fault-tolerant metric dimension of $J_4$, the graph with $3^4$ leaves minus 5 plus the added $s_i,r_i$ vertices. The paper predicts $\operatorname{ftdim}(J_4)\ge 3^3-4-1=22$, so a fault-tolerant resolving set of size 21 or less would refute the lower-bound construction. More directly, check whether any vertex other than a pair of leaves with ternary labels $0abc$ and $1abc$ resolves that pair; the proof of Theorem 2.8 asserts that only the two leaves and $s_1$ do.

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Extended reading notes

Core claim

The central discovery is that the largest possible fault-tolerant metric dimension among graphs of metric dimension $k$ grows like $3^k$ up to subexponential factors, i.e. $\lim_{k\to\infty}\frac{\log_3(\max_{G:\dim(G)=k}\operatorname{ftdim}(G))}{k}=1$. Toward this limit, the paper proves $\operatorname{ftdim}(G)\le \dim(G)(2+3^{\dim(G)-1})$ for every graph of order greater than 1, improving the long-standing base-5 upper bound, and constructs connected graphs $J_k$ with $\dim(J_k)=k$ and $\operatorname{ftdim}(J_k)\ge 3^{k-1}-k-1$, forcing the matching base-3 lower bound. The same machinery proves the analogous base-2 limit for fault-tolerant edge metric dimension and yields sharp extremal bounds for adjacency dimension and $k$-truncated metric dimension, together with characterizations of when the fault-tolerant values equal 2 or $n$.

Load-bearing premise

The lower-bound construction rests on the assertion that in $J_k$, a pair of leaves whose ternary labels agree except in the first digit (0 and 1) is resolved only by the two leaves themselves and by $s_1$; if any additional vertex also distinguished such a pair, the $3^{k-1}$ lower bound would collapse.

Editorial extensions

If this is right

For every graph $G$, a fault-tolerant resolving set of size at most $\dim(G)(2+3^{\dim(G)-1})$ always exists, and the exponential base 3 cannot be lowered because the graphs $J_k$ require roughly $3^{k-1}$ vertices in any fault-tolerant resolving set.
The analogous limit for edge metric dimension holds with base 2: the largest fault-tolerant edge metric dimension among graphs of edge metric dimension $k$ grows like $2^k$.
For $k$-truncated metric dimension with $k\ge 2$, the fault-tolerance base is again 3; for adjacency dimension, the 1-truncated case, the base is 2.
In every variant considered, a fault-tolerant resolving set of size 2 exists exactly when the ordinary resolving set has size 1, and the paper gives neighborhood conditions under which the fault-tolerant value equals the order $n$.
The maximum clique number of a graph of edge metric dimension at most $k$ equals the Erdős–Kleitman function $ek(k)$, so any improvement on $ek(k)$ transfers directly to that extremal graph problem.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The pattern in the paper suggests a general principle: for a metric-dimension variant whose maximum degree grows like $c^k$, the fault-tolerant version should have worst-case exponential base $c$; Section 9 asks exactly when this principle can fail, and the base-5-to-3 correction here shows why the principle needs proof rather than analogy.
The equivalence with the Erdős–Kleitman problem is two-way: any set family with more than about $(8/3)^{k/6}$ members having distinct pairwise unions would immediately produce edge-metric-dimension-$k$ graphs with larger cliques, and any upper-bound technique for such cliques would transfer back to set families.
A testable extension is to restrict to trees or bounded-degree graphs: the exponential lower-bound graphs are high-degree stars, so fault-tolerant metric dimension may grow only polynomially in $k$ on trees, and computing it for path and star families would show whether the $3^k$ phenomenon is driven by high degree.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's the take: the paper settles the exponential base for fault-tolerant metric dimension — it's 3, not the 5 from Hernando et al. or the 2 from Knor et al. That was a real open gap, and the resolution is clean. The same framework also gives base 2 for edge and adjacency dimension, base 3 for truncated dimension, and an equivalence between max clique with bounded edge metric dimension and the Erdős–Kleitman set-family problem.

The upper bound construction (closed neighborhood of a resolving set plus one extra vertex per landmark) is simple and works. The lower bound graph J_k is a smart ternary gadget; I checked the stress-test re-derivation and the counting in Theorem 2.8 is fine — exactly one size-3 cell loses two vertices, k−1 lose one, so |R| = 3^{k−1} − k, matching the theorem. The proof's intermediate 'at most k+1' is an off-by-one, but it's the harmless direction and the conclusion holds.

The soft spots are all presentational. The abstract claims ftdim(G) ≤ dim(G)(1+3^{dim−1}), but Theorem 2.7 proves dim(G)(2+3^{dim−1}). The abstract also says the lower bound is 3^{k−1}−k−1 while the theorem gives 3^{k−1}−k. J_1 is not connected, so the 'infinite family of connected graphs' needs k≥2 or a tweak. Section 8's construction says uncovered coordinates have distance 2; they can be ∞. That doesn't break the equivalence because ∞ is constant on the uncovered elements, but the text should say '2 or ∞.' There's also a typo in Theorem 2.8 where |S| should be |N|.

None of this changes the asymptotic result. The paper is honest about what it proves, the key limit follows from the stated bounds, and the dependence on prior work (including the first author's own papers) is legitimate — those are independent published tools.

For a reader: if you care about metric dimension variants or extremal set theory, this is worth your time. I'd bring it to reading group and I'd cite it. It deserves a proper referee; after the abstract and small fixes, it should be accepted.

Referee Report

2 major / 5 minor

Summary. The paper studies how large fault-tolerant versions of metric dimension can be relative to the original dimension. For standard metric dimension it proves ftdim(G) ≤ dim(G)(2+3^{dim(G)-1}) and constructs graphs J_k with dim(J_k)=k and ftdim(J_k) ≥ 3^{k-1}-k, yielding the limit lim_{k→∞} max_{G: dim(G)=k} log_3(ftdim(G))/k = 1. Analogous base-2 limits are proved for fault-tolerant edge metric dimension and adjacency dimension, and a base-3 limit is proved for k-truncated metric dimension for k ≥ 2. The paper also characterizes graphs with fault-tolerant dimension 2 or n for several variants, proves sharp degree bounds for vertices in resolving sets, and establishes an equivalence between the maximum clique number of graphs of edge metric dimension at most k and the Erdős–Kleitman problem on families with distinct pairwise unions.

Significance. If the results are correct, the paper closes a real gap by showing that the exponential base of fault-tolerant metric dimension is 3, improving on the previous base-5 upper bound and base-2 lower bound. The edge and adjacency analogues, and the truncated-metric extension, give a coherent picture in which the fault-tolerant exponential base matches the maximum-degree base. The constructive lower bounds are explicit and elementary, and the upper-bound method (closed neighborhood plus a bounded number of added vertices) is simple and likely to be reusable. The equivalence with the Erdős–Kleitman problem is a nice bridge between graph theory and extremal set theory. The paper's reliance on the first author's earlier theorems ([11], [12], [13]) is a standard use of established tools and does not appear circular.

major comments (2)

[Abstract; Theorem 2.7] The abstract states ftdim(G) ≤ dim(G)(1+3^{dim(G)-1}), but Theorem 2.7 proves only the weaker bound ftdim(G) ≤ dim(G)(2+3^{dim(G)-1}). The extra additive term arises because the proof adds one vertex u_s for each landmark s. Thus the abstract's advertised upper bound is not established by the manuscript, and the statement should be corrected to match the theorem (or the theorem must be strengthened). The asymptotic limit in Corollary 2.9 is unaffected by this correction.
[Theorem 2.8 proof (also Theorems 3.9 and 4.9)] The proof of Theorem 2.8 concludes that |T − {s1}| ≥ 3^{k-1} − k − 1 and then states the theorem's bound ftdim(J_k) ≥ 3^{k-1} − k. The inference appears to use |T| ≥ |T − {s1}| + 1, which is valid only if s1 ∈ T; the proof does not justify this. If s1 ∉ T, the same counting gives only |T| ≥ |R|. The gap is repairable: exactly one partition cell has size 1 and k−1 cells have size 2, so |R| = 3^{k-1} − k, and if s1 ∉ T then fault-tolerance forces both vertices of each full cell into T. As written, however, the proof is incomplete. The same 'remove the first landmark' issue appears in the proofs of Theorem 3.9 and Theorem 4.9, where it should be handled with the analogous case split.

minor comments (5)

[Theorem 2.8; Abstract] The statement that J_k is connected with dim(J_k)=k fails for k=1: J_1 is a disjoint union of two edges, and dim(J_1)=2. Since only k ≥ 2 is used for the limits, the theorem should be stated for k ≥ 2 or include a separate treatment of k=1.
[Theorem 2.8 proof] The sentence about r_i says the distance is 1 to both v and w when the common ith digit is 0 or 1; when the digit is 0, the distance is actually 2 via s_i. This does not affect the conclusion because the distances to v and w remain equal, but it should be corrected.
[Section 8, Theorem 8.1] In the forward direction of Theorem 8.1, the distance-vector formulas for clique edges and for edges {u_i, v_j} assume every coordinate not in the relevant union has value 2. If an index appears in no subset of the family F, the corresponding vertex u_t is isolated and the coordinate is ∞, not 2. This should be stated; the distinctness argument is unaffected because differences only occur at coordinates belonging to some subset.
[Theorem 7.4] The proof says that a landmark v has one coordinate equal to 0 and the rest positive; for disconnected graphs the other coordinates may be infinite. The upper bound still holds, but the sentence should allow ∞.
[Theorem 2.8 proof] The notation is confusing: the proof uses S both for the resolving set {s1,...,sk} and for the set of non-center vertices in the copy of K1,3^k when it writes '|S| = 3^k − k − 1'. Also, in the final paragraph 'T − {u1}' should be 'T − {s1}'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: central lower and upper bounds are independent; cited self-results are published lemmas, not definitions of the target quantities.

full rationale

The central derivation is not circular. The upper bound in Theorem 2.7 starts from an arbitrary resolving set S and builds S'' via closed neighborhoods, with the only external size estimate being the published degree bound |N(s)| ≤ 3^{|S|-1} from [12]; this bound is not a restatement of ftdim. The lower bound in Theorem 2.8 introduces a new graph J_k and argues directly that for each full ternary cell only the two vertices themselves and s1 resolve a 0/1 pair, forcing one vertex per cell into any fault-tolerant resolving set; that forcing argument is a structural computation, not a fitted input. The citations [11], [12], and [13] are prior published results used as lemmas, and they do not assume the theorem being proved. The Section 8 equivalence is an explicit two-way bijection between uniquely resolved clique edges and set families with distinct pairwise unions, so it is a genuine translation rather than a renaming of a known result. The same holds for the edge, adjacency, and truncated variants: upper bounds use closed neighborhoods plus published degree bounds, while lower bounds use explicit H_k, A_k, and J_k constructions with direct cell-counting arguments. No fitted parameter is renamed as a prediction, and no uniqueness theorem is imported from the authors' prior work to force a choice. Minor self-citations occur, but they are not load-bearing in the circular sense, so the score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted to data; the paper is a pure combinatorial proof. The load-bearing external inputs are several prior theorems on degree bounds, order bounds, and a family of extremal graphs, all cited and used as black boxes. No new physical or mathematical entities are postulated.

assumptions (5)

domain assumption Maximum degree of a vertex in a graph of metric dimension k is at most 3^k - 1 (Geneson et al. [12]).
Used in Theorem 2.7 to bound |N(s)| <= 3^(|S|-1) and in Theorem 2.8 to show dim(J_k) >= k. The paper does not prove this bound; it is a cited theorem.
domain assumption Any graph with adjacency dimension k has order at most k + 2^k (Geneson and Yi [13]).
Used in Theorem 4.8 to bound ftadim(G) <= adim(G) + 2^adim(G). Cited, not proved in the paper.
domain assumption Lemma 7.1 of [12]: For every graph H and positive integer k, there exists a graph of metric dimension k containing H iff D_k contains H.
Used in the proof of Theorem 7.2 to establish the exact count (2j+1)^k - 1 for graphs of metric dimension k. The lemma is cited from prior work.
domain assumption The graph H_k from [11] satisfies edim(H_k)=k and has vertices in its edge resolving set with degree 2^(k-1).
Used in Theorem 3.6 for sharpness of the degree bound and in Theorem 3.9 for the lower bound on ftedim. The properties are cited from [11].
standard math Standard graph metric facts: triangle inequality and the identity dist(s,x)=1+min over neighbors x' of s of dist(x',x) for a non-isolated vertex s.
Used throughout the upper-bound proofs in Sections 2, 3, and 5 to show that removing a landmark s still leaves enough information to distinguish vertices or edges.

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Cite this review

Pith. "Pith review of Fault tolerance for metric dimension and its variants." pith.science (2026). https://pith.science/paper/6BVT7D57

@misc{pith2026250202731,
  author       = {Pith},
  title        = {Pith review of: Fault tolerance for metric dimension and its variants},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6BVT7D57}},
  note         = {Machine review of arXiv:2502.02731}
}

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 Erd\H{o}s and Kleitman (1974) in extremal set theory.

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

[1] [11]

Geneson, Metric dimension and pattern avoidance in g raphs

J. Geneson, Metric dimension and pattern avoidance in g raphs. Discrete Applied Math- ematics 284 (2020) 1-7

work page 2020

[2] [12]

Geneson, S

J. Geneson, S. Kaustav, and A. Labelle. Extremal result s for graphs of bounded metric dimension. Discrete Applied Mathematics 309 (2022) 123-12 9

work page 2022

[3] [13]

Geneson, E

J. Geneson, E. Yi. Broadcast dimension of graphs. Austr alasian Journal of Combina- torics 83 (2022) 243-264

work page 2022

[4] [1]

Abrishami, M

G. Abrishami, M. A. Henning, M. Tavakoli, Local metric di mension for graphs with small clique numbers, Discrete Math. 345 (2022) Paper 11276 3

work page 2022

[5] [2]

Barrag´ an-Ram ´ ırez, J.A

G.A. Barrag´ an-Ram ´ ırez, J.A. Rodr ´ ıguez-Vel´ azquez, The local metric dimension of strong product graphs, Graphs Combin. 32 (2016) 1263–1278

2016

[6] [3]

Beardon, J.A

A.F. Beardon, J.A. Rodr ´ ıguez-Vel´ azquez, On the k-metric dimension of metric spaces, Ars Math. Contemp. 16 (2019) 25–38

work page 2019

[7] [4]

Chartrand, L

G. Chartrand, L. Eroh, M.A. Johnson, and O.R. Oellermann , Resolvability in graphs and the metric dimension of a graph, Discrete Applied Mathem atics 105 (2000), no. 1, 99–113

work page 2000

[8] [5]

Erd˝ os, D.J

P. Erd˝ os, D.J. Kleitman, Extremal problems among subse ts of a set. Disc. Math. 8 (1974) 281-294

work page 1974