Mutual visibility in Moore graphs and $(d,2)$-graphs with defect

Shikhi M; Tonny K B

arxiv: 2510.25858 · v2 · submitted 2025-10-29 · 🧮 math.CO

Mutual visibility in Moore graphs and (d,2)-graphs with defect

Tonny K B , Shikhi M This is my paper

Pith reviewed 2026-05-18 02:50 UTC · model grok-4.3

classification 🧮 math.CO

keywords mutual visibilityMoore graphsHoffman-Singleton graphinteger programminginduced matchinggraph invariantsdefect graphs

0 comments

The pith

The mutual-visibility number of the Hoffman-Singleton graph is exactly 20.

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

The paper derives algebraic conditions that bound the mutual-visibility number for (d,2)-graphs with non-negative defect. It then computes exact values for small d in the defect -2 case and an upper bound for d=5. For the Hoffman-Singleton graph, the authors prove an upper bound of 20 and use integer programming to show the bound is tight. This yields the corollary that the largest induced matching in the graph has size 10. A reader would care because the result fixes a new combinatorial invariant on one of the best-known Moore graphs and links it directly to matching theory.

Core claim

We derive algebraic conditions for the mutual-visibility number of (d,2)-graphs with non-negative defect. We determine this parameter for (d,2,-2)-graphs for d=3 and 4, and establish an upper bound for d=5. In the case of the Hoffman-Singleton graph we establish an upper bound of 20 for its mutual-visibility number and employ an integer programming approach to show that this bound is tight. As a corollary we deduce that the maximum size of an induced matching in the Hoffman-Singleton graph is 10.

What carries the argument

Algebraic conditions on the mutual-visibility number for (d,2)-graphs with defect, together with an integer programming formulation that finds the largest mutually visible set in the Hoffman-Singleton graph.

If this is right

The mutual-visibility number of the Hoffman-Singleton graph equals 20.
The Hoffman-Singleton graph has no induced matching larger than size 10.
The mutual-visibility number is fully determined for all (d,2,-2)-graphs when d equals 3 or 4.
Algebraic upper bounds hold for the mutual-visibility number of (d,2)-graphs with non-negative defect when d equals 5.

Where Pith is reading between the lines

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

The same integer-programming technique could be applied to other Moore graphs of diameter 2 to obtain exact mutual-visibility numbers.
The link between mutual visibility and induced matchings offers a new route to bounding matching numbers in vertex-transitive graphs of small diameter.
The derived algebraic conditions may extend to graphs with larger diameter or different defect values.

Load-bearing premise

The integer programming model correctly encodes all mutual-visibility constraints of the Hoffman-Singleton graph without omitted or relaxed conditions.

What would settle it

Exhibiting a set of 21 vertices in the Hoffman-Singleton graph in which every pair is mutually visible, or proving that the integer program has no feasible solution at objective value 21, would show the number is not 20.

read the original abstract

The concept of mutual visibility in a graph encodes combinatorial information about vertex subsets with prescribed visibility properties and serves as a useful algebraic invariant. In this paper, we derive algebraic conditions for the mutual-visibility number of $(d,2)$-graphs with non-negative defect. We then determine this parameter for $(d,2,-2)$-graphs for $d=3$ and $4$, and establish an upper bound for $d=5$. In the case $\delta=0$, that is, for Moore graphs of diameter $2$, we focus on the Hoffman-Singleton graph. We establish an upper bound of $20$ for its mutual-visibility number and subsequently employ an integer programming approach to show that this bound is tight. As a corollary, we deduce that the maximum size of an induced matching in the Hoffman--Singleton graph is $10$.

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.

Desk Editor's Note private letter to a colleague

The paper computes exact mutual-visibility numbers for (d,2,-2) graphs at d=3 and 4, then uses an algebraic bound plus IP to pin the Hoffman-Singleton case at 20 with a matching corollary.

read the letter

The main thing here is that the authors fill in some missing exact values for mutual visibility on small (d,2) graphs and settle the Hoffman-Singleton number at 20. They start with algebraic conditions that apply to (d,2)-graphs with non-negative defect, then handle the defect -2 cases for d=3 and 4 by direct determination. For the Moore graph case they derive an upper bound of 20 and tighten it with integer programming, which also gives the induced-matching size of 10 as a corollary. That computational step is the part that actually produces the new numbers rather than just restating known bounds. The algebraic setup looks consistent with prior work on these graphs and the IP check is presented as an independent verification, which keeps the circularity low. The results are concrete and limited in scope, which is fine for this kind of invariant. The obvious soft spot is the IP model itself. If the constraints do not fully capture every mutual-visibility requirement or if some linearization step is off, the feasible solution of size 20 could be invalid. The paper treats the encoding as correct, but that is the load-bearing claim and the one referees will examine most closely. The graphs are small enough that the formulation can be checked, yet the details matter. This is a paper for people already working on visibility parameters, cage graphs, or Moore graphs. A reader who wants precise numbers on these specific families will find it useful and can verify the claims without much extra work. It is narrow enough that it will not change the broader field, but the calculations are the sort that accumulate into useful tables. I would send it to peer review. The contribution is modest but the method is checkable and the claims are specific enough to be worth referee time.

Referee Report

1 major / 2 minor

Summary. The manuscript derives algebraic conditions for the mutual-visibility number of (d,2)-graphs with non-negative defect. It determines the exact value for (d,2,-2)-graphs when d=3 and d=4, and gives an upper bound for d=5. For the Hoffman-Singleton graph (a Moore graph of diameter 2), the authors establish an algebraic upper bound of 20 on the mutual-visibility number and use an integer programming formulation to prove that this bound is tight. As a corollary, they obtain that the maximum induced matching in the Hoffman-Singleton graph has size 10.

Significance. If the central claims hold, the work advances the algebraic and computational study of mutual-visibility parameters in highly symmetric graphs, including Moore graphs. The combination of derived algebraic conditions with an independent integer-programming verification for the Hoffman-Singleton graph, together with the induced-matching corollary, provides a concrete contribution to extremal graph theory and combinatorial invariants. The parameter-free algebraic upper bound and the computational confirmation are strengths that merit publication once the encoding details are fully verifiable.

major comments (1)

The integer-programming section for the Hoffman-Singleton graph: the claim that the IP establishes tightness of the algebraic upper bound of 20 (and hence the induced-matching corollary of size 10) depends on the model exactly encoding all pairwise mutual-visibility constraints derived from the (d,2)-graph setup. The manuscript should include the explicit constraint list, variable definitions, and linearization steps so that readers can confirm no constraints are omitted or relaxed; without this, a feasible solution of size 20 could be spurious.

minor comments (2)

Abstract: the phrase '(d,2)-graphs with defect' would benefit from a one-sentence parenthetical reminder of the defect parameter for readers outside the immediate subfield.
Notation section: ensure that the symbol for the mutual-visibility number is introduced before its first use in the algebraic conditions, and that it is consistently distinguished from the defect parameter.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their detailed and constructive report. The single major comment concerns the level of detail provided for the integer-programming verification in the Hoffman-Singleton case. We address this point directly below and will revise the manuscript to incorporate the requested information.

read point-by-point responses

Referee: The integer-programming section for the Hoffman-Singleton graph: the claim that the IP establishes tightness of the algebraic upper bound of 20 (and hence the induced-matching corollary of size 10) depends on the model exactly encoding all pairwise mutual-visibility constraints derived from the (d,2)-graph setup. The manuscript should include the explicit constraint list, variable definitions, and linearization steps so that readers can confirm no constraints are omitted or relaxed; without this, a feasible solution of size 20 could be spurious.

Authors: We agree that explicit documentation of the integer-programming model is necessary for independent verification. In the revised version we will add a dedicated subsection that (i) defines all decision variables (including binary indicators for vertex selection and auxiliary variables for pairwise visibility), (ii) lists every linear constraint that encodes the mutual-visibility condition derived from the (d,2)-graph algebraic setup, and (iii) details the linearization steps that convert the original quadratic visibility constraints into an equivalent integer linear program. These additions will make the encoding fully transparent and will allow readers to confirm that no constraints have been omitted or relaxed. We believe this revision will eliminate any ambiguity regarding the tightness result and the induced-matching corollary. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained via independent algebraic bound and computational verification

full rationale

The paper first derives algebraic conditions for mutual-visibility numbers in (d,2)-graphs with defect, then specializes to obtain an upper bound of 20 for the Hoffman-Singleton graph (a Moore graph of diameter 2). It subsequently applies an integer program whose feasible region is defined directly from the visibility constraints to exhibit a feasible solution of size 20, confirming tightness. This IP step is an external computational check on existence rather than a restatement or fitting of the algebraic bound; the induced-matching corollary follows from the verified set size. No equation reduces to a prior definition or fitted parameter by construction, no load-bearing self-citation chain is invoked, and the central claims remain independent of the verification method.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Paper relies on standard definitions from graph theory and extremal combinatorics; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)

standard math Standard definitions of graphs, Moore graphs, defect, and mutual visibility
Invoked throughout to set up the algebraic conditions and IP model.

pith-pipeline@v0.9.0 · 5669 in / 1150 out tokens · 36490 ms · 2026-05-18T02:50:41.403061+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Lemma 5 … G[S] has maximum degree at most 1 … induced matching … integer program (IP-MV) … μ(H)=20
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 20 … Jensen’s inequality … F''(e)>0 … s≤20

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

31 extracted references · 31 canonical work pages · 1 internal anchor

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[1] [1]

Aljohani, P

A. Aljohani, P. Poudel, and G. Sharma. Complete visitability for autonomous robots on graphs. InProceedings of the IEEE International Parallel and Distributed Processing Sym- posium (IPDPS), pages 733–742, 2018.doi:10.1109/IPDPS.2018.00083

work page doi:10.1109/ipdps.2018.00083 2018

[2] [2]

Alsaedi, J

R.J. Alsaedi, J. Gudmundsson, and A. van Renssen. The mutual visibility problem for fat robots.Algorithms and Data Structures. WADS 2023, 14079:15–28, 2023.doi:10.1007/ 978-3-031-38906-1_2

work page 2023

[3] [3]

Bhagat, P

S. Bhagat, P. Dey, and R. Ray. Collision-free linear time mutual visibility for asynchronous fat robots. InProceedings of the 25th International Conference on Distributed Computing and Networking, pages 84–93, 2024.doi:10.1145/3631461.3631544

work page doi:10.1145/3631461.3631544 2024

[4] [4]

Boruzanlı Ekinci, Cs

G. Boruzanlı Ekinci and Cs. Bujt´ as. Mutual-visibility problems in kneser and johnson graphs.Ars Mathematica Contemporanea, 2024.doi:10.26493/1855-3974.3344.4c8

work page doi:10.26493/1855-3974.3344.4c8 2024

[5] [5]

Breˇ sar, I.G

B. Breˇ sar and I.G. Yero. Lower (total) mutual-visibility number in graphs.Applied Math- ematics and Computation, 456:128411, 2024.doi:10.1016/j.amc.2023.128411

work page doi:10.1016/j.amc.2023.128411 2024

[6] [6]

Bujt´ as, S

Cs. Bujt´ as, S. Klavˇ zar, and J. Tian. Total mutual-visibility in hamming graphs.Opuscula Mathematica, 45:63–78, 2025.doi:10.7494/OpMath.2025.45.1.63

work page doi:10.7494/opmath.2025.45.1.63 2025

[7] [7]

Visibility polynomials, dual visibility spectrum, and characterization of total mutual-visibility sets.arXiv preprint, 2024

Csilla Bujt´ as, Sandi Klavˇ zar, and Jing Tian. Visibility polynomials, dual visibility spectrum, and characterization of total mutual-visibility sets.arXiv preprint, 2024.doi:10.48550/ arXiv.2412.03066

work page arXiv 2024

[8] [8]

Induced matchings.Discrete Applied Mathematics, 24(1–3):97–102, 1989

Kathie Cameron. Induced matchings.Discrete Applied Mathematics, 24(1–3):97–102, 1989. URL:https://doi.org/10.1016/0166-218X(92)90275-F,doi:10.1016/0166-218X(89) 90089-3

work page doi:10.1016/0166-218x(92)90275-f 1989

[9] [9]

Gary Chartrand, H´ ector H´ evia, and Robin J. Wilson. The ubiquitous petersen graph. Discrete Mathematics, 100:303–311, 1992.doi:10.1016/0012-365X(92)90649-Z. ON THE MUTUAL VISIBILITY OF SOME MOORE GRAPHS 13

work page doi:10.1016/0012-365x(92)90649-z 1992

[10] [10]

Cicerone, A

S. Cicerone, A. Di Fonso, G. Di Stefano, and A. Navarra. The geodesic mutual visibility problem for oblivious robots: The case of trees. InICDCN ’23: Proceedings of the 24th International Conference on Distributed Computing and Networking, pages 150–159, 2023. doi:10.1145/3571306.3571401

work page doi:10.1145/3571306.3571401 2023

[11] [11]

Cicerone, A

S. Cicerone, A. Di Fonso, G. Di Stefano, and A. Navarra. Time-optimal geodesic mu- tual visibility of robots on grids within minimum area. InStabilization Safety and Security of Distributed Systems(SSS 2023), volume 14310, page 385–399, 2023.doi: 10.1007/978-3-031-44274-2_29

work page doi:10.1007/978-3-031-44274-2_29 2023

[12] [12]

Cicerone, G

S. Cicerone, G. Di Stefano, L. Droˇ z ¯dek, J. Hedˇ zet, S. Klavˇ zar, and I.G. Yero. Variety of mutual-visibility problems in graphs.Theoretical Computer Science, 974:114096, 2023. doi:10.1016/j.tcs.2023.114096

work page doi:10.1016/j.tcs.2023.114096 2023

[13] [13]

Cicerone, G

S. Cicerone, G. Di Stefano, and S. Klavˇ zar. On the mutual-visibility in cartesian products and in triangle-free graphs.Applied Mathematics and Computation, 438:127619, 2023. doi:10.1016/j.amc.2022.127619

work page doi:10.1016/j.amc.2022.127619 2023

[14] [14]

2020, Results in Physics, 16, 102918, doi:10.1016/j

S. Cicerone, G. Di Stefano, S. Klavˇ zar, and I.G. Yero. Mutual-visibility problems on graphs of diameter two.European Journal of Combinatorics, 120:103995, 2024.doi:10.1016/j. ejc.2024.103995

work page doi:10.1016/j 2024

[15] [15]

Springer Berlin Heidelberg, 2012

Zdravko Cvetkovski.Convexity, Jensen’s Inequality. Springer Berlin Heidelberg, 2012. doi:10.1007/978-3-642-23792-8_7

work page doi:10.1007/978-3-642-23792-8_7 2012

[16] [16]

Di Luna, P

G.A. Di Luna, P. Flocchini, S.G. Chaudhuri, F. Poloni, N. Santoro, and G. Viglietta. Mutual visibility by luminous robots without collisions.Information and Computation, 254:392–418, 2017

work page 2017

[17] [17]

Addison-Wesley Series in Mathematics

Frank Harary.Graph Theory. Addison-Wesley Series in Mathematics. Addison-Wesley, Reading, MA, 1969

work page 1969

[18] [18]

Hoffman and Robert R

Alan J. Hoffman and Robert R. Singleton. On moore graphs with diameter 2.IBM Journal of Research and Development, 4(5):497–504, 1960

work page 1960

[19] [19]

Klavˇ zar, G

S. Irˇ siˇ c, S. Klavˇ zar, G. Rus, and J. Tuite. General position polynomials.Results in Mathe- matics, 79, 2024.doi:10.1007/s00025-024-02133-3

work page doi:10.1007/s00025-024-02133-3 2024

[20] [20]

Korˇ ze and A

D. Korˇ ze and A. Vesel. Mutual-visibility sets in cartesian products of paths and cycles. Results in Mathematics, 79:Article 116, 2024.doi:10.1007/s00025-024-02139-x

work page doi:10.1007/s00025-024-02139-x 2024

[21] [21]

Appl Math Comput

D. Korˇ ze and A. Vesel. Variety of mutual-visibility problems in hypercubes.Applied Math- ematics and Computation, 491:129218, 2025.doi:10.1016/j.amc.2024.129218

work page doi:10.1016/j.amc.2024.129218 2025

[22] [22]

Kuziak and J

D. Kuziak and J. A. Rodr´ ıguez-Vel´ azquez. Total mutual-visibility in graphs with emphasis on lexicographic and cartesian products.Bulletin of the Malaysian Mathematical Sciences Society, 46:197, 2023.doi:10.1007/s40840-022-01334-y

work page doi:10.1007/s40840-022-01334-y 2023

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