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arxiv: 2201.07799 · v5 · submitted 2022-01-19 · 🧮 math.GM

A Minimum Doubly Resolving Set and Strong Resolving Set for the Crystal Cubic Carbon

Ali Zafari , Saeid Alikhani This is my paper

Pith reviewed 2026-05-24 12:52 UTC · model grok-4.3

classification 🧮 math.GM
keywords crystal cubic carbonCCC(n)doubly resolving setstrong resolving setresolving setsgraph theoryminimum set size
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The pith

An alternative structural representation for crystal cubic carbon graphs CCC(n) is used to determine the minimum sizes of their doubly resolving sets and strong resolving sets.

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

The paper introduces an alternative structural representation for the crystal cubic carbon graph CCC(n). This representation is then used to calculate the minimum sizes of a doubly resolving set and a strong resolving set for the graph. Resolving sets have applications in chemistry, robot navigation, and pattern recognition, where they help uniquely identify locations or structures. A sympathetic reader would care because explicit minimum sizes for these sets in a carbon structure model could support more precise analysis in those fields.

Core claim

The paper introduces an alternative structural representation for the crystal cubic carbon CCC(n). Building on this model, the minimum sizes of both a doubly resolving set and a strong resolving set for CCC(n) are determined.

What carries the argument

The alternative structural representation of CCC(n) that captures adjacency and distance properties to compute resolving sets.

If this is right

  • The minimum size of a doubly resolving set for CCC(n) is now known explicitly for each n.
  • The minimum size of a strong resolving set for CCC(n) is now known explicitly for each n.
  • These sizes can be applied directly in any model that uses the crystal cubic carbon graph for identification tasks.

Where Pith is reading between the lines

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

  • The same representation might simplify computation of other metric dimension parameters on CCC(n).
  • If the sizes grow slowly with n, the graph family could support efficient location identification even at large scales.

Load-bearing premise

The alternative structural representation introduced in the paper correctly captures the adjacency and distance properties of the crystal cubic carbon graph CCC(n).

What would settle it

For a small n such as n=1, compute all pairwise distances in the actual CCC(1) graph and test whether the proposed minimum doubly resolving set distinguishes every pair of vertices under the doubly resolving condition.

Figures

Figures reproduced from arXiv: 2201.07799 by Ali Zafari, Saeid Alikhani.

Figure 3
Figure 3. Figure 3: Crystal cubic carbon CCC(2) 8 5 1 4 7 6 2 3 (x11 , 8)(2) (x11 , 7)(2) (x11 , 3)(2) (x11 , 4)(2) (x11 , 5)(2) (x11 , 6)(2) (x11 , 2)(2) (x11 , 1)(2) 2). Let n and k be fixed positive integers so that n ≥ 3, k ≥ 2 and [n] = {1, 2, ..., n}, also, let p be an integer so that 2 ≤ p ≤ k. In this section, we construct a class of graphs of order n + Σk p=2 n 2 (n − 1)p−2 , denoted by LCG(n, k) and recall that the … view at source ↗
Figure 4
Figure 4. Figure 4: Layer cycle graph LCG(5, 3) 3 2 1 5 4 (x11 , 5)(2) (x11 , 4)(2) (x11 , 3)(2) (x11 , 2)(2) (x11 , 1)(2) (x11 , 1)(3) (x11 , 5)(3) (x11 , 4)(3) (x11 , 3)(3) (x11 , 2)(3) (x12 , 4)(3) (x12 , 3)(3) (x12 , 2)(3) (x12 , 1)(3) (x12 , 5)(3) (x13 , 4)(3) (x13 , 3)(3) (x13 , 2)(3) (x13 , 1)(3) (x13 , 5)(3) (x14 , 4)(3) (x14 , 3)(3) (x14 , 2)(3) (x14 , 1)(3) (x14 , 5)(3) 2. Main Results Theorem 2.1. Consider the crys… view at source ↗
read the original abstract

The task of identifying resolving sets has been extensively studied due to its wide relevance in fields such as chemistry, robot navigation, combinatorial optimization, pattern recognition, and image processing. These applications have helped motivate and establish the theoretical foundations of the subject. Notably, problems of this type are generally known to be NP-hard. This study introduces an alternative structural representation for the crystal cubic carbon \( CCC(n) \). Building on this model, we determine the minimum sizes of both a doubly resolving set and a strong resolving set for $CCC(n)$.

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

1 major / 0 minor

Summary. The paper introduces an alternative structural representation for the crystal cubic carbon graph CCC(n) and, building on this model, determines the minimum sizes of both a doubly resolving set and a strong resolving set for CCC(n).

Significance. If the alternative representation faithfully preserves all adjacencies and distances of the original CCC(n) graph, the explicit minimum sizes would constitute concrete, computable results for two metric-dimension variants on a family of chemical graphs, with potential relevance to the listed application areas. The manuscript supplies no machine-checked proofs, reproducible code, or parameter-free derivations.

major comments (1)
  1. [Introduction / model definition (as described in the abstract)] The central claim rests on the unverified assertion that the introduced alternative structural representation preserves all adjacencies and distances of the standard crystal cubic carbon graph. No isomorphism proof, distance-table comparison for small n, or embedding into the lattice description is supplied to confirm faithfulness; any mismatch would invalidate the subsequent resolving-set calculations.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comment on our manuscript. The primary concern is the absence of explicit verification that the alternative structural representation preserves the adjacencies and distances of the standard CCC(n) graph. We address this point below and commit to strengthening the manuscript accordingly.

read point-by-point responses

  1. Referee: [Introduction / model definition (as described in the abstract)] The central claim rests on the unverified assertion that the introduced alternative structural representation preserves all adjacencies and distances of the standard crystal cubic carbon graph. No isomorphism proof, distance-table comparison for small n, or embedding into the lattice description is supplied to confirm faithfulness; any mismatch would invalidate the subsequent resolving-set calculations.

    Authors: We acknowledge that the original submission did not contain an explicit isomorphism proof, distance-table verification for small n, or a lattice embedding to confirm that the alternative model preserves all adjacencies and distances. In the revised manuscript we will add a self-contained section establishing the equivalence: we will construct an explicit bijection between the vertex sets that preserves adjacency, prove that distances are identical under this mapping, and include a table comparing distances for n=1,2,3 together with a description of the embedding into the underlying lattice. These additions will directly address the concern and validate the subsequent metric-dimension results. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation applies standard resolving-set definitions to the newly introduced graph model

full rationale

The paper introduces an alternative structural representation for CCC(n) and then determines minimum doubly resolving set and strong resolving set sizes by direct calculation on that model. No self-definitional quantities appear, no parameters are fitted to data and relabeled as predictions, and no load-bearing self-citations or uniqueness theorems are invoked. The steps consist of standard graph-theoretic arguments applied to the defined adjacency and distance relations, rendering the derivation self-contained against external benchmarks. The correctness of the representation is an assumption whose verification lies outside the derivation chain itself.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities are described.

pith-pipeline@v0.9.0 · 5612 in / 956 out tokens · 47846 ms · 2026-05-24T12:52:38.599122+00:00 · methodology

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Reference graph

Works this paper leans on

31 extracted references · 31 canonical work pages

  1. [1]

    Godsil and G

    C. Godsil and G. Royle, Algebraic Graph Theory, Springer, New Y ork, NY , USA, 2001

    work page 2001

  2. [2]

    P . J. Cameron and J. H. V an Lint, Designs Graphs, Codes and their Links , Cambridge University Press, Cambridge, UK, 1991

    work page 1991

  3. [3]

    Resolvability in graphs and the metric dimension of a graph ,

    G. Chartrand, L. Eroh, M. A. Johnson, O. R. Oellermann, “Resolvability in graphs and the metric dimension of a graph ,” Discrete Applied Mathematics, vol. 105, pp. 99-113, 2000

    work page 2000

  4. [4]

    Khuller, B

    S. Khuller, B. Raghavachari and A. Rosenfeld, Localization in graphs, Technical Report CS-TR-3326, University of Maryland at College Park, 1994

    work page 1994

  5. [5]

    On metric generators of graphs ,

    A. Seb¨ o and E. Tannier, “On metric generators of graphs ,” Math. Oper . Res, vol.29, pp. 383-393, 2004

    work page 2004

  6. [6]

    Leaves of trees ,

    P . J. Slater, “Leaves of trees ,” in Proceedings of the 6th Southeastern Conference on Combin atorics, Graph theory and Computing , Boca Raton, FL, USA, pp. 549-559, 1975

    work page 1975

  7. [7]

    Molecular description of carbon graphite and crystal cubi c carbon structures,

    A. Q. Baig, M. Imran, W. Khalid, and M. Naeem, “Molecular description of carbon graphite and crystal cubi c carbon structures,” Canadian Journal of Chemistry, vol. 95(6), pp. 674-686, 2017

    work page 2017

  8. [8]

    On two problems of information theory ,

    P . Erd¨ os and P . R. Alfr´ ed,“On two problems of information theory ,” Magyar tudomanyos akademia matematikai kutato intezetene k kozle- menyei, vol. 8, pp. 229-243, 1963

    work page 1963

  9. [9]

    Landmarks in graphs,

    S. Khuller, B. Raghavachari, and A. Rosenfeld, “Landmarks in graphs,” Discrete Applied Mathematics, vol. 70, no. 3, pp. 217-229, 1996

    work page 1996

  10. [10]

    On a combinatory detection problem I. I. Magyar Tud. Akad. M at. Kutat oaccute,

    B. Lindstr¨ om, “On a combinatory detection problem I. I. Magyar Tud. Akad. M at. Kutat oaccute,” Int Kazi, vol. 9, pp. 195-207, 1964

    work page 1964

  11. [11]

    Application of Resolvability Technique to Investigate the Di fferent Polyphenyl Structures for Polymer Industry ,

    M. F. Nadeem, M. Hassan, M. Azeem, S. U. D. Khan, M. R. Shai k, M. A. F. Sharaf, A. Abdelgawad, and E. M. Awwad, “Application of Resolvability Technique to Investigate the Di fferent Polyphenyl Structures for Polymer Industry ,” Journal of Chemistry , vol. 2021, pp. 1-8, 2021

    work page 2021

  12. [12]

    On k-dimensional graphs and their bases ,

    P . S, Buczkowski, G. Chartrand, C. Poisson, and P . Zhang , “On k-dimensional graphs and their bases ,” Periodica Math Hung, vol. 46(1), pp. 9-15, 2003

    work page 2003

  13. [13]

    On the metric dimension of Mobius ladders ,

    M. Ali, G. Ali, M. Imran, A. Q. Baig, and M. K. Shafiq, “On the metric dimension of Mobius ladders ,” Ars Comb, vol. 105, pp. 403-410, 2012

    work page 2012

  14. [14]

    Structure-activity maps for visualizing the graph variab les arising in drug design ,

    M. A. Johnson, “Structure-activity maps for visualizing the graph variab les arising in drug design ,” Journal of Biopharmaceutical Statistics, vol. 3, pp. 203-236, 1993

    work page 1993

  15. [15]

    Computing Metric Dimension of Certain Families of Toeplit z Graphs ,

    J.-B. Liu, M. F. Nadeem, H. M. A. Siddiqui, and W. Nazir, “Computing Metric Dimension of Certain Families of Toeplit z Graphs ,” IEEE Access, vol. 7, pp. 126734-126741, 2019

    work page 2019

  16. [16]

    Metric dimension, minimal doubly resolving sets and stron g metric dimension for Jellyfiish graph and Cocktail party graph,

    J.-B. Liu, A. Zafari, and H. Zarei, “Metric dimension, minimal doubly resolving sets and stron g metric dimension for Jellyfiish graph and Cocktail party graph,” Complexity, vol. 2020, pp. 1-7, 2020

    work page 2020

  17. [17]

    Some resolving parameters in a class of Cayley graphs ,

    J.-B. Liu and A. Zafari, “Some resolving parameters in a class of Cayley graphs ,” Journal of Mathematics, vol. 2022, pp. 1-5, 2022

    work page 2022

  18. [18]

    Computing the metric dimension of wheel related graphs ,

    H. M. A. Siddiqui, M. Imran, “Computing the metric dimension of wheel related graphs ,” Applied Mathematics and Computation , vol. 242, pp. 624-632, 2014

    work page 2014

  19. [19]

    Computing the metric dimension of the categorial product o f some graphs ,

    T. V etr´ ık and A. Ahmad, “Computing the metric dimension of the categorial product o f some graphs ,” International Journal of Computer Mathematics, vol. 94(2), pp. 363-371, 2017

    work page 2017

  20. [20]

    On the metric dimension of a graph ,

    F. Harary and R. A. Melter, “On the metric dimension of a graph ,” Combinatoria, vol. 2, pp. 191-195, 1976

    work page 1976

  21. [21]

    On the metric dimension of Cartesian products of graphs,

    J. C´ aceres, C. Hernando, M. Mora, I. M. Pelayo, M. L. Pue rtas, C. Serra, and D. R. Wood, “On the metric dimension of Cartesian products of graphs,” SIAM Journal on Discrete Mathematics, vol. 21, pp. 423-441, 2007

    work page 2007

  22. [22]

    Minimal doubly resolving sets of necklace graph ,

    A. Ahmad, M. Baˇ ca, and S. Sultan, “Minimal doubly resolving sets of necklace graph ,” Math. Reports, vol. 20(70) 2, pp. 123-129, 2018

    work page 2018

  23. [23]

    Computing minimal doubly resolving sets of graphs ,

    J. Kratica, M. ˇCangalovi´ c and V . Kovaˇ cevi´ c-Vujˇ ci´ c,“Computing minimal doubly resolving sets of graphs ,” Comput. Oper . Res, vol. 36(7), pp. 2149-2159, 2009

    work page 2009

  24. [24]

    Computing minimal doubly resolving sets and the strong met ric dimension of the layer Sun graph and the Line Graph of the Layer Sun Graph ,

    J.-B. Liu and A. Zafari, “Computing minimal doubly resolving sets and the strong met ric dimension of the layer Sun graph and the Line Graph of the Layer Sun Graph ,” Complexity, vol. 2020, pp. 1-8, 2020

    work page 2020

  25. [25]

    Fault-Tolerant Resolvability in Some Classes of Line Grap hs,

    X. Guo, M. Faheem, Z. Zahid, W. Nazeer, and J. Li, “Fault-Tolerant Resolvability in Some Classes of Line Grap hs,” Mathematical Problems in Engineering, vol. 2020, pp. 1-8, 2020

    work page 2020

  26. [26]

    On the strong metric dimension of corona product graphs and join graphs,

    D. Kuziak, Ismael G. Y ero, and J. A. Rodr´ ıguez-V el´ azquez, “On the strong metric dimension of corona product graphs and join graphs,” Discrete Applied Mathematics, vol. 161, pp. 1022-1027, 2013

    work page 2013

  27. [27]

    The strong metric dimension of graphs and digraphs ,

    O. R. Oellermann and J. Peters-Fransen, “The strong metric dimension of graphs and digraphs ,” Discrete Applied Mathematics, vol. 155, pp. 356-364, 2007

    work page 2007

  28. [28]

    On the strong metric dimension of Cartesian and direct prod ucts of graphs,

    J. A. Rodr´ ıguez-V el´ azquez, I. G. Y ero, D. Kuziak, andO. R. Oellermann, “On the strong metric dimension of Cartesian and direct prod ucts of graphs,” Discrete Mathematics, vol. 335, pp. 8-19, 2014

    work page 2014

  29. [29]

    Topological Characterization of Carbon Graphite and Crys tal Cubic Carbon Structures,

    W. Gao, M. K. Siddiqui, M. Naeem, and N. A. Rehman, “Topological Characterization of Carbon Graphite and Crys tal Cubic Carbon Structures,” Molecules, vol. 22(9):1496, pp. 1-12, 2017

    work page 2017

  30. [30]

    Molecular properties of carbon crystal cubic structures,

    H. Y ang, M. Naeem, and M. K. Siddiqui, “Molecular properties of carbon crystal cubic structures,” Open Chemistry, vol. 18(1), pp. 339–346, 2020

    work page 2020

  31. [31]

    Metric Dimension of Crystal Cubic Carbon Structure ,

    X. Zhang and M. Naeem, “Metric Dimension of Crystal Cubic Carbon Structure ,” Journal of Mathematics, vol. 2021, pp. 1-8, 2021. 9

    work page 2021