On the Double Roman Domination Number of Generalized Sierpinski Graphs

Anu V.; Aparna Lakshmanan S.

REVIEW 3 major objections 3 minor 12 references

On the Double Roman Domination Number of Generalized Sierpinski Graphs

T0 review · 3 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The double Roman domination number of the generalized Sierpiński graph $S(K_n,2)$ is exactly $3n-1$ for complete graphs, and a two-sided bound controls the parameter for every base graph.

desk verdict A credible exact value for double Roman domination on S(K_n,2), with a lower-bound proof that needs a missing appeal to a known lemma plus some inequality typo cleanup. read the letter →

arxiv 1908.06858 v1 pith:DIMG24KV submitted 2019-08-19 math.CO

Anu V. , Aparna Lakshmanan S. This is my paper · ORCID

classification math.CO MSC 05C6905C76

keywords doubleRomandominationSierpińskigraphsgeneralizednumbercomplete

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

This paper studies how the double Roman domination number changes when a graph is expanded into a generalized Sierpiński graph $S(G,t)$, a recursively built graph whose vertices are length-$t$ words over the vertices of $G$. It proves a two-sided estimate for every graph $G$: the double Roman domination number of $S(G,t)$ lies between $n^{t-2}\alpha(G)\gamma_{dR}(G)$ and $n^{t-2}(n\gamma_{dR}(G)-|V_3|-|D_3|)$, where $\alpha(G)$ is the independence number and $V_3$, $D_3$ describe a cheapest double Roman dominating function of $G$. For complete graphs with $t=2$, the bound becomes exact: $\gamma_{dR}(S(K_n,2)) = 3n-1$. This matters because exact values for domination-type parameters on Sierpiński graphs are rare, and the proof constructs explicit labelings that achieve them.

What carries the argument

The load-bearing object is the generalized Sierpiński graph $S(G,t)$, whose vertices are length-$t$ words over the vertex set of $G$ and whose edges swap adjacent letters according to edges of $G$. For the lower bound, the paper partitions the vertex set into $n^{t-1}$ sets $V_{wi}=\{wij: j\in V(G)\}$, each inducing a copy of $G$; choosing a maximum independent set in $G$ produces copies with no edges among them and no common neighbors, so each copy is argued to need its own $\gamma_{dR}(G)$ weight. For the upper bound, a three-step reweighting construction clones the optimal function on every copy, moves 3-valued vertices onto diagonal words, and then frees non-isolated diagonal 3-vertices by setting them to 0, yielding a valid function of the stated smaller weight.

What would settle it

Run an exact minimum-weight search for a double Roman dominating function on $S(K_4,2)$; Theorem 3.2 predicts a weight of 11, so any feasible assignment of weight 10 or less would refute the exact-value claim, and a parallel search on a small non-complete base graph such as $S(P_3,2)$ would test the general lower bound.

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Extended reading notes

Core claim

The paper's central claim is that $\gamma_{dR}(S(G,t))$ is controlled by local data of $G$ alone: for a graph $G$ of order $n$, with independence number $\alpha(G)$, and with a cheapest double Roman dominating function whose vertex classes have sizes $|V_3|$ and $|D_3|$ (where $D_3$ is the set of non-isolated vertices among those assigned 3), the double Roman domination number of $S(G,t)$ satisfies the two-sided inequality in Theorem 2.2. When $G=K_n$ and $t=2$, the bound is tight and gives $\gamma_{dR}(S(K_n,2)) = 3n-1$; the same section proves the analogous Roman domination value $\gamma_R(S(K_n,2)) = 2n-1$. The exact-value proof shows that any cheapest function must pay 2 at one extreme vertex and 3 at each of the other $n-1$ opposite vertices, while the upper-bound construction exhibits a function of exactly that weight.

Load-bearing premise

The lower-bound proof assumes that each of the $n^{t-1}$ disjoint copies of $G$ inside $S(G,t)$ must be double-Roman-dominated from its own vertices, contributing at least $\gamma_{dR}(G)$ independently, even though vertices in neighboring copies are adjacent to those copies and could in principle help dominate them.

Editorial extensions

If this is right

For complete graphs, the exact value $\gamma_{dR}(S(K_n,2)) = 3n-1$ fixes the per-vertex cost of the second-level Sierpiński construction at roughly 3 per base vertex.
For ordinary Roman domination, the matching exact value $\gamma_R(S(K_n,2)) = 2n-1$ gives a clean comparison point between the two domination parameters.
For any base graph, the double Roman domination number of every $S(G,t)$ lies in an interval determined only by $n$, $\alpha(G)$, $\gamma_{dR}(G)$, and the extremal structure of a cheapest function.
The upper-bound proof constructs an explicit valid labeling, so it provides an algorithmic way to produce a double Roman dominating function of the stated weight, not merely an existence statement.

Reading between the lines

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

Editorial inference: the lower bound's disjointness assumption could be probed on small non-complete base graphs; exact search on, say, $S(P_3,2)$ or $S(C_4,2)$ would either confirm the bound or reveal a case where cross-copy domination lowers the true value below the formula.
Editorial inference: the same three-step reweighting scheme may apply to other domination-type parameters (for instance total Roman or independent Roman domination), because the proof relies only on open-neighborhood conditions shared by those parameters.
Editorial inference: the two-sided estimate suggests that $\gamma_{dR}(S(G,t))$ grows like a constant times $n^t$, and a natural next step would be to determine whether the coefficient $n^{t-2}$ appearing in the theorem is sharp beyond the complete-graph case.

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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

The one genuinely new thing here is the exact value γdR(S(K_n,2)) = 3n−1, plus the general bounds for S(G,t). The upper bound comes from an explicit construction and is fine; the lower-bound idea is also basically right. The proof of Theorem 3.2, however, has a real gap: it lets f be any γdR-function and never invokes Proposition 1.1, which says we can take a γdR-function with no vertex of value 1. Without that assumption, the conclusion that the unique 2 in the exceptional copy must be at the extreme vertex does not follow. Adding “By Proposition 1.1, assume V1=∅” repairs the argument, and then the rest of the case analysis is sound. So the central claim holds up, but only after a fix the authors did not make explicit.

The soft spots in the rest of the paper are milder. Theorem 2.1 uses a lower-bound argument for independent copies that assumes each copy needs a disjoint contribution of γR(G); the proof asserts this from disjoint open neighborhoods but does not fully rule out external support. The reader flagged this, and it is a fair concern, but it is not load-bearing for the exact-value theorem, because Theorem 3.2’s lower bound is self-contained once Proposition 1.1 is added. There are also several inequality direction typos ('>' where '≥' is meant), which are annoying but not substantive.

The citation pattern is clean: the relevant prior work on Roman domination of Sierpinski graphs and on double Roman domination is cited, and the new result genuinely extends that literature. It is a modest, incremental result, not a breakthrough, but it is a legitimate one.

Who is this for? Someone working on domination-type parameters on Sierpinski graphs or on double Roman domination specifically. It is not going to change anyone’s research program, but it is the kind of paper that a serious referee could read in an hour and return with a short list of fixes.

Recommendation: send it to peer review. It deserves referee time, not a desk reject. The authors should be asked to fix the inequality typos, prove the per-copy lower bound more carefully or flag it as a lemma, and explicitly invoke Proposition 1.1 in Theorem 3.2.

Referee Report

3 major / 3 minor

Summary. The paper studies the double Roman domination number of generalized Sierpiński graphs. It states a lower bound for the Roman domination number of S(G,t) (Theorem 2.1), a two-sided bound for the double Roman domination number of S(G,t) (Theorem 2.2), and then specializes to complete graphs, proving γR(S(Kn,2)) = 2n−1 and the main claim γdR(S(Kn,2)) = 3n−1. The upper bound in Theorem 2.2 is constructive, obtained by reassigning weights in three steps, and gives an explicit double Roman dominating function of weight 3n−1 on S(Kn,2). The lower bound for the exact value is a case analysis on the rows Gi of S(Kn,2).

Significance. The claimed exact value γdR(S(Kn,2)) = 3n−1 is natural and, with the repairs described below, the argument is very likely correct. The upper-bound construction in Theorem 2.2 is explicit and yields a clean equality case: one extreme vertex has value 2 and all swapped partners have value 3. The proof strategy is elementary and checkable, and the paper correctly relies on standard cited results rather than introducing circular assumptions. However, the lower-bound proof of the main theorem omits a necessary appeal to Proposition 1.1, contains incorrect strict inequalities, and the general lower bound in Theorem 2.1 is not justified as written. These are local, fixable gaps rather than a fundamental flaw, but they preclude acceptance in the present form.

major comments (3)

[Section 3, proof of Theorem 3.2] The proof lets f be any γdR-function and concludes, for the exceptional row Gi0, that the unique vertex of value 2 must be the extreme vertex ui0ui0. This conclusion is only valid when V1 is empty; otherwise f(ui0ui0)=1 together with one value-2 neighbor would already double-Roman-dominate the extreme, and the later claim that every ui0uj (j≠i0) is dominated only through ujui0 fails for vertices ui0uj that themselves have value 1. The proof should begin with 'By Proposition 1.1, choose a γdR-function with V1=∅.' With that sentence inserted, the remaining case analysis is sound.
[Section 3, proof of Theorem 3.2] The strict inequalities in this proof are false as written. If every row Gi contains a vertex of value 3 or two vertices of value 2, the total weight is at least 3n, not strictly greater than 3n; the function with every extreme vertex equal to 3 has weight exactly 3n. Similarly, the final line f(V)>2+3(n−1) should be f(V)≥2+3(n−1), since the upper-bound construction attains equality. Replacing '>' by '≥' preserves the contradiction with the upper bound 3n−1 and is necessary for deriving the exact value. Also, the sentence 'it is optimal to assign the value 3 to ujui0' is an assertion rather than a proof; the intended justification is that if ujui0=2, then ujuj forces a further vertex of value at least 2 in Gj, making the row cost at least 4, whereas ujui0=3 costs exactly 3.
[Section 2, Theorem 2.1 and left inequality of Theorem 2.2] The proof asserts the lower bound from the facts that the copies Vwi have no edges between them and have pairwise disjoint open neighborhoods. This does not by itself imply that the total weight is at least n^{t−2}α(G)γR(G): a vertex outside a copy can still dominate a vertex inside that copy, so one must prove a charging lemma showing that each copy together with the external vertices that neighbor it contributes at least γR(G) (or γdR(G), for Theorem 2.2). As written, the per-copy cost claim is unsupported. In addition, the strict inequality in Theorem 2.1 should be ≥, not >. These flaws do not affect the upper-bound construction used in Theorem 3.2, but they affect the paper's claimed general bound.

minor comments (3)

[Section 3, Theorem 3.1] The same type of gap appears in the proof of the Roman domination result: the sentence 'there exists at least one Gi0 which contains exactly one vertex in V1' requires an exchange argument, and the assertion 'to Roman dominate ui0uj, ujui0∈V2' presumes that ui0uj is not itself assigned 1. A short minimality argument should be supplied.
[Section 2, Theorem 2.2] The statement says 'for any integer t>2', but the application to S(Kn,2) in Theorem 3.2 uses t=2; the condition should read t≥2. Theorem 2.1 should also explicitly state t≥2 so that the words w∈V^{t−2} exist.
[Throughout] There are numerous typographical and formatting issues: the name 'Sierpi´nski' appears with an inverted accent, expressions such as 'nt−2' and 'V t' are inconsistently superscripted, and several inequalities are written with '>' where the surrounding argument requires '≥'. A careful proofreading pass is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity; the exact-value proof is self-contained and does not reduce to its inputs or to load-bearing self-citations.

full rationale

The paper's central result, gamma_dR(S(K_n,2)) = 3n-1, is derived from definitions plus external results, not from a fitted parameter or a self-citation chain. The upper bound applies Theorem 2.2 to K_n with a genuine construction g2 and the known values gamma_dR(K_n)=3, |V3|=1, |D3|=0; it does not assume the target value. The lower bound is a direct case analysis on an arbitrary gamma_dR-function, and the only imported structural fact, Proposition 1.1 (one may take V1=empty), comes from the independent source [6], not from the authors' own work. Theorem 1.2 from [11] is likewise external and parameter-free. The authors' self-citations [3,4] appear only in the introduction as background and are not used in the proofs. There are proof-rigor gaps: Theorem 3.2 implicitly uses Proposition 1.1 to exclude assigning value 1 to the extreme vertex ui0ui0 without invoking it, and Theorem 2.1's lower-bound proof contains a questionable disjoint-neighborhood claim inherited by Theorem 2.2. These are correctness concerns, not circularity: no equation is shown to be its own input, no fitted value is relabeled as a prediction, and no load-bearing premise is supported solely by a self-citation. Honest non-finding is therefore appropriate.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No free parameters or invented entities. The paper relies on standard definitions, a cited proposition (V1 can be assumed empty), a cited theorem on Roman domination of Sierpinski graphs, and an unproved structural disjointness/cost claim for the lower bound.

assumptions (3)

standard math For any graph G, there exists a γdR-function with V1=∅ (Proposition 1.1 from [6]).
Invoked to restrict attention to functions f=(V0,V2,V3) in the bounds and exact-value proof.
domain assumption In S(G,t), for an independent set V' of G, the copies V_wi for i∈V' are pairwise non-adjacent and have pairwise disjoint open neighborhoods.
Used to derive the lower bound in Theorem 2.1/2.2; asserted from the edge definition without a detailed proof of the cost implication.
domain assumption The minimum weight needed to double-Roman-dominate each independent copy V_wi, even with the help of its external neighbors, is at least γdR(G).
This is the unproved leap in the lower-bound proof: disjointness of neighborhoods is shown, but the per-copy cost lower bound is stated as immediate.

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Pith. "Pith review of On the Double Roman Domination Number of Generalized Sierpinski Graphs." pith.science (2026). https://pith.science/paper/DIMG24KV

@misc{pith2026190806858,
  author       = {Pith},
  title        = {Pith review of: On the Double Roman Domination Number of Generalized Sierpinski Graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DIMG24KV}},
  note         = {Machine review of arXiv:1908.06858}
}

abstract

In this paper, we study the double Roman domination number of generalized Sierpi\'{n}ski graphs $S(G,t)$. More precisely, we obtain a bound for the double Roman domination number of $S(G, t)$. We also find the exact value of $\gamma_{dR}(S(K_{n}, 2))$.

Figures

Figures reproduced from arXiv: 1908.06858 by the authors.

**Figure 1.** A graph G and S(G, 2). Theorem 1.2. [11] For any integers n > 2 and t > 1, γR(S(Kn, t)) 6 ( 2n t+n−1 n+1 , t even, 2(n t+1) n+1 , t odd. 2 Bounds on the Double Roman Domination Number First we prove a lower bound for γR(S(G, t)). Theorem 2.1. For any graph G of order n, γR(S(G, t)) > n t−2α(G)γR(G), where α(G) is the independence number of G. Proof. Let V 0 ⊆ V be an independent set of cardinality α(G). For any w ∈ … view at source ↗

**Figure 2.** S(G, 3) for the graph G in [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

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Reference graph

Works this paper leans on

12 extracted references · 9 canonical work pages

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

H. A. Ahangar, M. A. Henning, V. Samodivkin, I. G. Yero, Total Roman Domination in Graphs , Appl. Anal. Discrete Math., 10 (2016), 501-517

2016

[2] [2]

Amjadi, S

J. Amjadi, S. Nazari-Moghaddam, S. M. Sheikholeslami, L. Volkmann, An Upper Bound on the Double Roman Domination Number , J. Comb. Optim., (to appear)

work page

[3] [3]

Math., 244 (2018), 198-204

Anu V., Aparna Lakshmanan S., Double Roman Domination Number , Dis- crete Appl. Math., 244 (2018), 198-204

2018

[4] [4]

Anu V., Aparna Lakshmanan S., Impact of Some Graph Operations on Double Roman Domination Number, (Submitted), (2018)

work page 2018

[5] [5]

Balakrishnan, K

R. Balakrishnan, K. Ranganathan, A Text Book of Graph Theory , Springer, New York, (1999)

work page 1999

[6] [6]

R. A. Beeler, T. W. Haynes, S. T. Hedetniemi, Double Roman Domination , Discrete Appl. Math., 211(1) (2016), 23-29

2016

[7] [7]

Gravier, M

S. Gravier, M. Kovˇ se, A. Parreau, Generalized Sierpi´ nski Graphs , in: Posters at EuroComb’11, R´ enyi Institute, Budapest, URL http://www.renyi.hu/conferences/ec11/posters/parreau.pdf, (Website visited last on 21.12.2018), (2011)

work page 2011

[8] [8]

G. Hao, X. Chen, L. Volkmann, Double Roman Domination in Digraphs, Bull. Malays. Math. Sci. Soc., (to appear)

work page