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arxiv: 2409.12199 · v1 · pith:A226A6NZnew · submitted 2024-09-09 · 💻 cs.DM · math.CO

Fault Tolerant Metric Dimensions of Leafless Cacti Graphs with Application in Supply Chain Management

Tauseef Asif , Ghulam Haidar , Faisal Yousafzai , Murad Ul Islam Khan , Qaisar Khan , Rakea Fatima This is my paper

Pith reviewed 2026-05-23 21:20 UTC · model grok-4.3

classification 💻 cs.DM math.CO
keywords fault tolerant metric dimensionleafless cacti graphsbicyclic graphsresolving setsmetric basissupply chain managementgraph decomposition
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The pith

The fault tolerant metric dimension of bicyclic graphs of type I and II is always 4, and leafless cacti graphs have this dimension expressed in terms of their inner and outer cycles.

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

The paper first proves that bicyclic graphs of type I and type II each have fault tolerant metric dimension exactly 4. These results then provide the basis for determining the fault tolerant metric dimension of leafless cacti graphs as a function of the number of inner cycles and outer cycles present in the graph. A resolving set distinguishes all vertices by distances, and a fault tolerant version remains distinguishing even after any single member is removed. The authors apply the findings to a supply chain example involving distribution centers.

Core claim

Bicyclic graphs of type I and type II have fault tolerant metric dimension 4. Leafless cacti graphs admit a calculation of their fault tolerant metric dimension directly from the counts of inner cycles and outer cycles, obtained by extending the bicyclic case analysis.

What carries the argument

Decomposition of leafless cacti graphs into inner and outer cycles that supports construction of minimal fault-tolerant resolving sets by case analysis on those cycles.

If this is right

  • Bicyclic graphs of type I always have fault tolerant metric dimension 4.
  • Bicyclic graphs of type II always have fault tolerant metric dimension 4.
  • The fault tolerant metric dimension of any leafless cacti graph is determined by its inner cycle count and outer cycle count.
  • In supply chain models represented by leafless cacti graphs, the minimal number of centers needed for fault-tolerant location equals the fault tolerant metric dimension of the graph.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The cycle-count formula may support polynomial-time computation of the dimension for any leafless cacti graph once the cycles are identified.
  • Similar decompositions could be tested on other cactus families or graphs with bounded treewidth to see if fault tolerant dimension remains expressible by local cycle features.
  • In network design problems, the result suggests that adding redundant location points scales predictably with the number of cycles rather than total vertices.

Load-bearing premise

Leafless cacti graphs admit a canonical decomposition into inner and outer cycles such that minimal fault-tolerant resolving sets can be constructed and verified by exhaustive case analysis on those cycles without additional vertices required in any configuration.

What would settle it

A leafless cacti graph whose fault tolerant metric dimension cannot be expressed solely as a function of its inner cycle count and outer cycle count, or that deviates from the value obtained by extending the bicyclic results.

Figures

Figures reproduced from arXiv: 2409.12199 by Faisal Yousafzai, Ghulam Haidar, Murad Ul Islam Khan, Qaisar Khan, Rakea Fatima, Tauseef Asif.

Figure 2.1
Figure 2.1. Figure 2.1: bicyclic graph of type-I Note that the total number of vertices is + − 1, while total number of edges in + . Also, it is worthwile to mention that the vertices of are indexed in an anti-clockwise direction, while vertices of are indexed clockwise. These graphs are also called “infinity-graphs” [28]. II. ,, obtained after joining two disjoint cycles, and , by a path ( ≥ 1) of length . Let us consider the … view at source ↗
Figure 2.2
Figure 2.2. Figure 2.2: bicyclic graph of type-II We see that the path attaches the vertex of to + of . Note that usually a path of length has + 1 vertices but here, the number of vertices on the path are − 1 due to the inclusion of vertices and +. Moreover, the vertices of are indexed anticlockwise, while that of are indexed clockwise. These graphs are also known as “Kayak-Paddle graph” [29]. In fact, Infinity-graph and Kayak … view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: Relationship of common vertex = + to metric basis of base bicyclic graph , of type-I. Without loss of generality, let ∈ and ∈ . If any of , are middle labeled vertices of their respective cycles, i.e., ≃ 2 or ≃ + [PITH_FULL_IMAGE:figures/full_fig_p004_3_1.png] view at source ↗
Figure 3.2
Figure 3.2. Figure 3.2: Relationship of n [PITH_FULL_IMAGE:figures/full_fig_p005_3_2.png] view at source ↗
Figure 3.3
Figure 3.3. Figure 3.3: , ∈ for base bicyclic graph , of type-I for odd , Now, (, ) = (, ) + (, ) and d(, ) = (, ) + (, ) implies that (, ) , (, ). Hence resolves the vertices and . Case 2. When , ∈ . Since , ∈ and are equidistant from ⌊ [PITH_FULL_IMAGE:figures/full_fig_p006_3_3.png] view at source ↗
Figure 3.4
Figure 3.4. Figure 3.4: , ∈ for base bicyclic graph , of type-I for odd , and on through Since is an odd cycle, we can easily conclude that || , |′ |. Since can be any vertex of , without loss of generality, we suppose that lies on . Following two possibilities arise for the placement of [PITH_FULL_IMAGE:figures/full_fig_p006_3_4.png] view at source ↗
Figure 3.5
Figure 3.5. Figure 3.5: , ∈ for base bicyclic graph , of type-I for odd , and on -path through +⌊ 2 ⌋ . If = +⌊ 2 ⌋ , we can easily see that resolves and by using the facts || , |′ | and (, ) = (, ). Next, we consider the cases when , +⌊ 2 ⌋ . II-(i): If the shortest -paths passes through +⌊ 2 ⌋ , we get ( , ) =  [PITH_FULL_IMAGE:figures/full_fig_p007_3_5.png] view at source ↗
Figure 3.6
Figure 3.6. Figure 3.6: Different occurrences of and equidistant from [PITH_FULL_IMAGE:figures/full_fig_p008_3_6.png] view at source ↗
Figure 3.7
Figure 3.7. Figure 3.7: Possibilities of elements of from (). Elements of are represented as red colored vertices in the above figure. One can see that in case , the two red vertices don’t resolve , of which are equidistant from . In case ℬ, without loss of generality, we assume that ∈ is from along with a vertex from . One can again see that the vertices , equidistant from + can not be resolved by this . This concludes our res… view at source ↗
Figure 3.8
Figure 3.8. Figure 3.8: Possibilities of Positions of , in ,,. In case B, since lies on the smallest path, we can easily conclude that ( , ) , ( , ). In case , since , are equidistant from , they are equidistant from +. Since is odd, no matter which ∈ we consider, we get ( , ) , ( , ). Case can be further divided into two cases. − 1 where , both lie in the same half of and case − 2, where they lie in different halves of . When … view at source ↗
Figure 5.1
Figure 5.1. Figure 5.1: Cacti Graph without Leaves Now, we are ready to extend the concept of fault tolerant metric generator to cacti graphs without leaves. Lemma 5.1. Let be a cactus graph without leaves then [PITH_FULL_IMAGE:figures/full_fig_p012_5_1.png] view at source ↗
Figure 5.2
Figure 5.2. Figure 5.2: figure 5.2 [PITH_FULL_IMAGE:figures/full_fig_p012_5_2.png] view at source ↗
Figure 5.2
Figure 5.2. Figure 5.2: Fault Tolerant Metric Generator of an Outer Cycle [PITH_FULL_IMAGE:figures/full_fig_p013_5_2.png] view at source ↗
Figure 5.3
Figure 5.3. Figure 5.3: Odd Inner Cycle in Cacti Graph Without Leaves. Let and be the corresponding outer cycles of and , respectively. Since and are both outer cycles, we have [PITH_FULL_IMAGE:figures/full_fig_p014_5_3.png] view at source ↗
Figure 5.4
Figure 5.4. Figure 5.4: Even Inner Cycle in Cacti Graph Without Leaves Having Unequal Paths Between Two Common Vertices [PITH_FULL_IMAGE:figures/full_fig_p014_5_4.png] view at source ↗
Figure 5.5
Figure 5.5. Figure 5.5: Even Inner Cycle in Cacti Graph Without Leaves Having Equal Paths Between Two Common Vertices. Let us now consider the vertices ′ [PITH_FULL_IMAGE:figures/full_fig_p015_5_5.png] view at source ↗
Figure 5.6
Figure 5.6. Figure 5.6: Even Inner Cycle in Leafless Cactus Graph with |()| = 2 and Having Antipodal Vertices. Now, any two vertices which are not resolved by , are resolved by ′ [PITH_FULL_IMAGE:figures/full_fig_p015_5_6.png] view at source ↗
Figure 5.7
Figure 5.7. Figure 5.7: Types of Inner Paths in a Leafless Cactus Graph. figure that in both types, there are always at least two outer cycles, say and , corresponding to the end points of the path . By Lemma 5.2, both these cycles contribute two vertices to the fault tolerant metric dimension of . We can now easily conclude that all the vertices of are fault tolerantly resolved by the union of fault tolerant metric generators … view at source ↗
Figure 6.1
Figure 6.1. Figure 6.1: Supply Chain Network as Leafless Cactus Graph This figure illustrates that the three main warehouses are all interconnected, so that if a warehouse has a shortage, the other two can overcome that. The warehouse 2 not only serves goods to retail outlets 6, 7, but also delivers them to the smaller warehouse 1. On the other hand, the warehouse 3 serves three smaller warehouses 2, 3, 4, which in turn serve t… view at source ↗
read the original abstract

A resolving set for a simple graph $G$ is a subset of vertex set of $G$ such that it distinguishes all vertices of $G$ using the shortest distance from this subset. This subset is a metric basis if it is the smallest set with this property. A resolving set is a fault tolerant resolving set if the removal of any vertex from the subset still leaves it a resolving set. The smallest set satisfying this property is the fault tolerant metric basis, and the cardinality of this set is termed as fault tolerant metric dimension of $G$, denoted by $\beta'(G)$. In this article, we determine the fault tolerant metric dimension of bicyclic graphs of type-I and II and show that it is always $4$ for both types of graphs. We then use these results to form our basis to consider leafless cacti graphs, and calculate their fault tolerant metric dimensions in terms of \textit{inner cycles} and \textit{outer cycles}. We then consider a detailed real world example of supply and distribution center management, and discuss the application of fault tolerant metric dimension in such a scenario. We also briefly discuss some other scenarios where leafless cacti graphs can be used to model real world problems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The paper claims that the fault-tolerant metric dimension β'(G) equals 4 for all bicyclic graphs of type I and type II. It then extends the result to leafless cacti graphs by expressing β' in terms of the numbers of inner and outer cycles, and illustrates the concept with a supply-chain example modeled by such graphs.

Significance. If the claimed closed-form values hold, the work would supply explicit, easily computable fault-tolerant resolving sets for two infinite families of graphs that arise in network modeling. The supply-chain application shows a concrete setting in which distance-based location under single-vertex failure is relevant.

major comments (2)
  1. [Abstract] Abstract: the central claim that β'(G)=4 for every bicyclic graph of type I and II is asserted without any proof, case analysis, or verification that candidate sets remain resolving after the removal of any single vertex. Because the subsequent leafless-cactus formulas rest on these bicyclic results, the absence of supporting derivations is load-bearing for the entire contribution.
  2. [leafless cacti section] The extension to leafless cacti (described after the bicyclic case) presupposes that every such graph admits a decomposition into type-I/II bicyclic blocks plus inner/outer cycles such that a fault-tolerant resolving set of size determined solely by the cycle counts can always be assembled. The manuscript provides no argument that this decomposition covers arbitrary attachment patterns (multiple cycles at one vertex, chains of blocks, etc.) or that no extra vertices are required to resolve distance ties once a resolver fails.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable comments. We address each major comment below and will incorporate revisions to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that β'(G)=4 for every bicyclic graph of type I and II is asserted without any proof, case analysis, or verification that candidate sets remain resolving after the removal of any single vertex. Because the subsequent leafless-cactus formulas rest on these bicyclic results, the absence of supporting derivations is load-bearing for the entire contribution.

    Authors: We agree that the abstract presents the result concisely. The full manuscript includes sections dedicated to proving that β'(G) = 4 for bicyclic graphs of type I and II, with case analyses based on the structure of these graphs. However, to make the supporting derivations more explicit and to verify the fault-tolerance property in detail, we will expand the case analysis in the revised version, including explicit checks that the proposed sets remain resolving after any single vertex removal. This will ensure the bicyclic results are self-contained before extending to leafless cacti. revision: yes

  2. Referee: [leafless cacti section] The extension to leafless cacti (described after the bicyclic case) presupposes that every such graph admits a decomposition into type-I/II bicyclic blocks plus inner/outer cycles such that a fault-tolerant resolving set of size determined solely by the cycle counts can always be assembled. The manuscript provides no argument that this decomposition covers arbitrary attachment patterns (multiple cycles at one vertex, chains of blocks, etc.) or that no extra vertices are required to resolve distance ties once a resolver fails.

    Authors: We acknowledge the need for a more rigorous justification of the decomposition for leafless cacti graphs. In the revised manuscript, we will add a dedicated subsection that formally describes the decomposition into bicyclic blocks and cycles, proves that it covers arbitrary attachment patterns including multiple cycles at vertices and chains of blocks, and demonstrates that the fault-tolerant resolving set size, determined by the numbers of inner and outer cycles, suffices without requiring additional vertices. This will include arguments showing that distance ties are resolved even after a resolver failure. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper determines fault-tolerant metric dimension for bicyclic graphs (always 4) via direct combinatorial arguments, then extends the result to leafless cacti by expressing it in terms of inner/outer cycle counts. No equations, parameters, or quantities are defined in terms of the quantities they are claimed to predict. No self-citations are invoked as load-bearing premises for uniqueness or ansatz choices. The central claims rest on case analysis of cycle structures rather than any self-referential reduction or fitted-input renaming. This is the expected non-finding for a purely combinatorial graph-theory paper whose results are stated as explicit determinations.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard definition of graph distance and resolving sets together with structural assumptions about the two graph families; no numerical parameters are fitted and no new entities are postulated.

axioms (2)
  • standard math Shortest-path distance in an undirected graph forms a metric that distinguishes vertices via unique distance vectors.
    This is the foundational property used to define resolving sets and fault-tolerant resolving sets.
  • domain assumption Bicyclic graphs of type I and II and leafless cacti graphs possess the cycle decompositions described in the paper.
    The formulas and the constant-4 claim are derived by case analysis on these specific cycle structures.

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Reference graph

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