Pith. sign in

REVIEW 3 major objections 5 minor 20 references

Italian Domination and Perfect Italian Domination on Sierpinski Graphs

T0 review · 3 major / 5 minor · reviewed 2026-08-27 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read For each Sierpiński graph $S(K_n,t)$ with $n,t\ge 3$, both the Italian and perfect Italian domination numbers equal $n^{t-2}(2n-2)$.

desk verdict Plausible exact values and a clean construction, but the main proof has an incomplete case analysis and an unproved jump from t=3 to all t; deserves a refereed major revision. read the letter →

arxiv 2009.09202 v1 pith:24KTWFFQ submitted 2020-09-19 math.CO

Jismy Varghese , Anu V , Aparna Lakshmanan S This is my paper · ORCID
classification math.CO MSC 05C6905C76
keywords ItaliandominationperfectSierpińskigraphsnumberRoman{2}-dominationrecursivegraphconstructionexactformula
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

The paper determines the Italian domination number of the Sierpiński graphs $S(K_n,t)$ exactly: for $n\ge 3$ and $t\ge 3$ it is $n^{t-2}(2n-2)$. It also shows that the stricter perfect Italian domination number, which requires every zero-weight vertex to see total neighbour-weight exactly $2$, takes this same value for all $t\ge 3$, with $t=2$ giving the separate value $2n-1$. The result matters because it gives a complete closed-form answer for two domination parameters on a recursively built family, and it shows that the perfect condition costs nothing on these graphs beyond depth $2$.

What carries the argument

The load-bearing object is the recursive decomposition of $S(K_n,t)$: it is built from $n$ copies of $S(K_n,t-1)$ by adding an edge between the extreme vertices $xyy\cdots y$ and $yxx\cdots x$. The base case $t=3$ is analysed copy by copy: every minimum Italian dominating function has the canonical shape in which each embedded $S_i(K_n,2)$ has weight $2n-2$, exactly two of its $K_n$ subcopies receive total weight $1$, and the remaining $n-2$ subcopies receive total weight $2$. The canonical shape creates a cyclic dependency between a copy and its neighbours, and the paper's corollary assumes this pattern multiplies unchanged through the $n^{t-3}$ levels of recursion.

What would settle it

Compute, by exhaustive search, the minimum weight of an Italian dominating function on the small case $S(K_4,4)$: the paper's formula predicts $4^{2}(6)=96$. The $t=3$ lower-bound argument does not explicitly rule out a cheaper function in which vertices in neighbouring copies cooperate across the edges joining embedded copies, so finding any Italian dominating function of weight below $96$ would refute Corollary 3.3.

Watch

Extended reading notes

Core claim

The central claim is that, for $n\ge 3$ and $t\ge 3$, $\gamma_I(S(K_n,t)) = n^{t-2}(2n-2)$, and the perfect Italian domination number $\gamma^p_I(S(K_n,t))$ equals the same number for $t\ge 3$, with both parameters equal to $2n-1$ when $t=2$. The proof constructs an Italian dominating function that assigns weight $1$ to exactly two vertices in each copy of $K_n$ inside each copy of $S(K_n,2)$, leaves every other vertex at weight $0$, and checks that each zero-weight vertex sees exactly two weight-$1$ neighbours, so the function is automatically perfect. The lower-bound argument for $t=3$ shows that every embedded copy of $S(K_n,2)$ in a minimum function must carry weight at least $2n-2$, with exactly two of its $K_n$ subcopies of total weight $1$ and the rest of total weight $2$; the corollary then multiplies this per-copy weight over $n^{t-3}$ copies.

Load-bearing premise

The step from $t=3$ to all larger $t$ assumes that no minimum Italian dominating function of the whole graph can save weight by letting vertices in one embedded copy help dominate zero-weight vertices in another embedded copy beyond the one external connection already analysed; the paper states this conclusion but does not prove the absence of such cooperation.

Editorial extensions

If this is right

  • For every $n\ge 3$ and $t\ge 3$, $\gamma_I(S(K_n,t)) = n^{t-2}(2n-2)$, so each extra level of recursion multiplies the domination cost by $n$.
  • For $t\ge 3$, $\gamma^p_I(S(K_n,t)) = \gamma_I(S(K_n,t))$, so the perfect-Italian 'exactly $2$' constraint imposes no additional weight on Sierpiński graphs.
  • For $t=2$, both parameters equal $2n-1$, so depth $2$ is the only small case with a different value.
  • At depth $3$, every minimum function assigns weight $0$ to all extreme vertices and total weight $2n-2$ to each embedded $S(K_n,2)$ copy, giving a rigid local structure.

Reading between the lines

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

  • The equality of the Italian and perfect Italian numbers for all depths $t\ge 3$ suggests that the perfect condition is automatically reachable within the optimal weight on this family; testing other recursively defined graph families would show whether this is a general phenomenon or a special feature of Sierpiński graphs.
  • The unproved induction from $t=3$ to larger $t$ is directly checkable: for $n=3$, $t=4$ the formula predicts $36$, and a small computer search for a cheaper dominating function would either confirm or refute the corollary.
  • If the formula holds, the explicit weight-$1$-only construction gives a ready-made upper bound for other domination-type parameters that only require a total neighbour weight of at least $2$ on zero-weight vertices.
Share X Bluesky LinkedIn Reddit HN

Formalized claims in Lean

  1. Claim #1: The central claim is that, for $n\ge 3$ and $t\ge 3$, $\gamma_I(S(K_n,t)) = n^{t-2}(2n-2)$, and the perfect Italian domination number $\gamma^p_I(S(K_n,t))$ equals the same number for $t\ge 3$, with both parameters equal to $2n-1$ when $t=2$. The proof constructs an Italian dominating function that assigns weight $1$ to exactly two vertices in each copy of $K_n$ inside each copy of $S(K_n,2)$, lea

Signed reviews

No signed human review yet.

Request a human review

A listed scientist reviews the paper for a fee and the review publishes here regardless of verdict. See the reviewers or get listed.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper studies the Italian domination number γI and the perfect Italian domination number γpI of Sierpiński graphs S(K_n,t). The main results are claimed to be exact formulas: γI(S(K_n,t)) = 2n−1 for t=2, and γI(S(K_n,t)) = n^{t−2}(2n−2) for n≥3 and t≥3; the same values are asserted for the perfect Italian domination number. The proof for t=3 is a construction of an IDF of weight 2n(n−1) followed by a lower-bound argument claiming that each of the n embedded copies of S(K_n,2) must have weight at least 2n−2. A remark then asserts a structural description of all minimum functions on S(K_n,3), and a corollary generalizes the formula to all t≥3 without a detailed proof. In Section 4 the t=2 perfect case is verified by an explicit weighting, and the t≥3 perfect case is concluded from the Italian case.

Significance. If the formulas are correct, the paper supplies exact values for two domination parameters on Sierpiński graphs and shows a possibly surprising coincidence γI = γpI for all depths t≥3. The upper-bound constructions are explicit and the t=2 perfect-Italian verification is elementary and sound. The main weakness is that the t>3 generalization and parts of the structural remark are underproved; the paper is therefore conditionally interesting but not yet complete. The stress-test concern about the unsupported induction in Corollary 3.3 lands: the manuscript text does not provide an argument that inter-copy connections cannot lower the total weight.

major comments (3)
  1. [§3, Theorem 3.2, lower-bound case analysis] The proof that every S_p(K_n,2) copy has weight at least 2n−2 reduces the assumption f(S_p(K_n,2)) ≤ 2n−3 to exactly two distributions of subcopy weights, namely (0,1,2,...,2) and (1,1,1,2,...,2). This reduction is valid only if every K_n-subcopy has weight in {0,1,2}; the proof nowhere rules out subcopy weights ≥3. For example, for n=5 the distribution (3,0,0,2,2) sums to 7 = 2n−3 and is not covered by either case. Since this enumeration is the basis for the claimed lower bound, the proof of Theorem 3.2 is incomplete as written.
  2. [§3, Remark and Corollary 3.3] Corollary 3.3 is the load-bearing step for all t>3, but it is not proved. The preceding remark is about γI-functions of S(K_n,3) as a standalone graph; it does not analyze interactions between the n^{t−3} embedded copies of S(K_n,3) in S(K_n,t). The copies are joined precisely at extreme-vertex edges, so a boundary vertex of one copy could in principle help dominate vertices in another copy and lower the total weight below n^{t−3}·(2n(n−1)) = n^{t−2}(2n−2). The sentence 'From the above remark we can arrive at the following corollary' does not supply the required induction or a lemma excluding such inter-copy savings. As it stands, both Corollary 3.3 and the t≥3 part of Theorem 4.1 are unsupported.
  3. [§3, Remark, structural claims] The remark asserts that in any γI-function of S(K_n,3) the extreme vertices have weight 0 and that within each S_i(K_n,2) exactly two K_n-subcopies have weight 1 and all others have weight 2. These assertions are not established by the preceding proof. The discussion of the extreme vertex only treats the case of weight 1, not weight 2, and the phrases 'this process is cyclically repeated' and 'will in turn increase the total weight' are not formal arguments. Moreover, the proof of Theorem 3.2 only rules out certain distributions for copies of total weight ≤ 2n−3; it does not determine the subcopy distribution at the minimum weight 2n−2. Since Corollary 3.3 relies on this structure, the remark needs a complete proof.
minor comments (5)
  1. [§4, Theorem 4.1, t=2] In the t=2 paragraph, after proving γpI(S(K_n,2)) ≤ 2n−1, the text uses γI ≤ γpI and γI = 2n−1 to conclude the equality, but writes 'Therefore, γpI(S(K_n,2)) ≤ 2n−1' twice. The lower bound γpI ≥ 2n−1 should be stated explicitly.
  2. [§4, Theorem 4.1, t≥3] The sentence 'In the proof of Theorem 3.2 we have defined an Italian dominating function with the property...' refers to a function on S(K_n,3), not on S(K_n,t). The analogue for S(K_n,t) should be exhibited, or at least it should be explicitly said that the construction is applied to each embedded S(K_n,3) copy and that the exact-neighbor property is preserved.
  3. [§2 and §4] For n=2, S(K_n,t) is a path on 2^t vertices, so the notation 'P_{2t}' should be corrected to 'P_{2^t}'.
  4. [§5, Problem 1] The inequality quoted from [14] uses the symbols V2 and E2, which are not defined in this paper; please define them or rephrase the bound.
  5. [§3, Theorem 3.2, construction] In the definition of the upper-bound function f, the indices i−1 and i+1 should be specified as modulo n, and the exceptional cases for j should be stated unambiguously.

Circularity Check

1 steps flagged · score 4.0 of 10

A self-cited communicated result supplies the t=2 perfect-Italian lower bound, but the central t≥3 formulas are derived independently and are not circular.

  1. self citation load bearing [Section 4, Theorem 4.1, Case 1 (t=2), relying on Theorem 3.1 and [14]]
    "Theorem 3.1. [14] The Italian domination number of the Sierpinski graph S(Kn, 2) is 2n−1. ... We know that γI(S(Kn, 2)) = 2n−1 and γI(S(Kn, 2)) ≤ γp I (S(Kn, 2)). Therefore, γp I (S(Kn, 2)) ≤ 2n−1. Hence γp I (S(Kn, 2)) = 2n−1."

    The lower bound γp_I(S(Kn,2)) ≥ 2n−1 is obtained only by combining the inequality γI ≤ γp_I with the equality γI(S(Kn,2)) = 2n−1, and the only support given for that equality is Theorem 3.1, which cites [14], a 'Communicated' paper by two of the present authors. The paper gives no independent derivation of γI(S(Kn,2)) = 2n−1, so the t=2 branch of the perfect-Italian claim is imported from the authors' own prior work rather than proved here. This does not affect the t≥3 formulas.

full rationale

The central derivation for t≥3 is self-contained: Theorem 3.2 explicitly constructs an IDF of weight 2n(n−1) on S(Kn,3) and proves a matching lower bound by a case analysis on the n copies of S(Kn,2); no fitted parameter or externally imported number is used. The perfect-Italian upper bound for t≥3 is presented as the same explicit assignment from Theorem 3.2, and the lower bound follows from γI ≤ γp_I together with Corollary 3.3, so that branch is internally derived. Corollary 3.3, however, is asserted from the structural remark about S(Kn,3) without an explicit induction showing that inter-copy connections cannot lower the weight in S(Kn,t); that is a proof gap (a correctness or completeness issue), not a circularity, because the corollary is not an input to its own proof. Similarly, the incomplete case analysis in the reverse direction of Theorem 3.2 (e.g., omitting configurations with a K_n-subcopy of weight 3 compensated by a weight-0 subcopy) is a proof gap, not a circular reduction. The only genuine circularity-adjacent step is the t=2 perfect-Italian lower bound, which leans on the authors' own communicated paper [14] for γI(S(Kn,2)) = 2n−1. Since that self-citation is load-bearing only for the t=2 subcase and not for the central t≥3 result, the overall circularity is limited.

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

The paper does not fit any parameters; the formulas are exact. The main unstated dependencies are the unpublished self-cited t=2 result and the unproved propagation of the t=3 structure to higher t.

assumptions (5)
  • standard math S(K_n,t) can be partitioned into n^{t-3} copies of S(K_n,3), with cross-copy edges only between extreme vertices of the copies.
    This follows from the recursive construction in Section 2, but the paper does not spell out the t>3 decomposition; it is needed for Corollary 3.3 and Theorem 4.1.
  • domain assumption gamma_I(S(K_n,2)) = 2n-1 for n at least 3.
    Stated as Theorem 3.1 and cited to the authors' own communicated paper [14]; used as the lower bound for the perfect Italian number when t=2.
  • ad hoc to paper In every gamma_I-function of S(K_n,3), extreme vertices have weight 0.
    Asserted in the remark after Theorem 3.2 with a brief justification; it is used to argue that cross-copy help cannot reduce the weight when t>3.
  • ad hoc to paper The t=3 weight pattern can be imposed independently on each embedded S(K_n,3) copy of S(K_n,t) to yield a valid (perfect) Italian dominating function for t>3.
    The paper does not explicitly construct or verify the PID for t>3; Theorem 4.1 Case 2 says 'In the proof of Theorem 3.2 we have defined...' without writing the extended assignment.
  • domain assumption Path formulas gamma_I(P_m) = ceil((m+1)/2) and gamma^p_I(P_m) = ceil((m+1)/2) handle the n=2 case.
    Cited to [1] and [17]; the n=2 case is not proved in this paper.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Italian Domination and Perfect Italian Domination on Sierpinski Graphs." pith.science (2026). https://pith.science/paper/24KTWFFQ

@misc{pith2026200909202,
  author       = {Pith},
  title        = {Pith review of: Italian Domination and Perfect Italian Domination on Sierpinski Graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/24KTWFFQ}},
  note         = {Machine review of arXiv:2009.09202}
}
abstract

An Italian dominating function (IDF) of a graph G is a function $ f: V(G) \rightarrow \{0,1,2\} $ satisfying the condition that for every $ v\in V $ with $ f(v) = 0$, $\sum_{ u\in N(v)} f(u) \geq 2. $ The weight of an IDF on $G$ is the sum $ f(V)= \sum_{v\in V}f(v) $ and the Italian domination number, $ \gamma_I(G) $, is the minimum weight of an IDF. An IDF is a perfect Italian dominating function (PID) on $G$, if for every vertex $ v \in V(G) $ with $ f(v) = 0 $ the total weight assigned by $f$ to the neighbours of $ v $ is exactly 2, i.e., all the neighbours of $u$ are assigned the weight 0 by $f$ except for exactly one vertex $v$ for which $ f(v) = 2 $ or for exactly two vertices $v$ and $w$ for which $ f(v) = f(w) = 1 $. The weight of a PID- function is $f(V)=\sum_{u \in V(G)}f(u)$. The perfect Italian domination number of $G$, denoted by $ \gamma^{p}_{I}(G), $ is the minimum weight of a PID-function of $G$. In this paper we obtain the Italian domination number and perfect Italian domination number of Sierpi\'{n}ski graphs.

Figures

Figures reproduced from arXiv: 2009.09202 by the authors.

Figure 1
Figure 1. Sierpi´nski graphs S(K5, t), t = 1, 2. the survey paper [12] appeared in 2017. For any graph theoretic terminology and notations not mentioned here, the readers may refer to [3]. 3 Main Results In this section, the exact value of the Italian domination number of the Sierpi´nski graph, γI (S(Kn, t)) is given. For n = 2, S(Kn, t) = P2 t and in [1] it has been proved that γI (Pn) = d n+1 2 e. For t = 2 and n ≥ 2 we alr… view at source ↗
Figure 2
Figure 2. Sierpi´nski graph S(K5, 3). • {{vivjvk, vivkvj} : i, j, k ∈ {1, 2, ..., n}, j 6= k}. • {{vivjvk, vivjvl} : i, j, k, l ∈ {1, 2, ..., n}, k 6= l}. Let Si(Kn, 2) denote the i th copy of S(Kn, 2) in S(Kn, 3) and Sij (Kn) denote the j th copy of Kn in Si(Kn, 2) for i, j = 1, 2, ..., n. Define an Italian dominating function on S(Kn, 3) as follows. f(v) =    1; v = vivjvi−1, i, j ∈ {1, 2, ..., n}, and v = vivjvi+1, i,… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

20 extracted references · 18 canonical work pages

  1. [14]

    Jismy, S

    V. Jismy, S. Aparna Lakshmanan, Italian Domination on Mycielskian and Sierpi´ nski Graphs,Communicated

    work page

  2. [1]

    Alizadeh, H

    F. Alizadeh, H. R. Maimani, L. P. Majd, M. R. Parsa, Roman{2}-domination in Graphs and Graph Products , Unpublished. 8

    work page

  3. [2]

    Anu V., Aparna Lakshmanan S., The Double Roman Domination Num- ber of Generalized Sierpi´ nski Graphs, Discrete Math., Algorithms and Appl., https://doi.org/10.1142/S1793830920500470 (accepted)

    work page doi:10.1142/s1793830920500470

  4. [3]

    Balakrishnan, K

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

    1999

  5. [4]

    Chellai, T

    M. Chellai, T. W. Haynes, S. T. Hedetniemi, A. A. McRae, Roman{2}- domination, Discrete Appl. Math., 204 (2016), 22-28

    work page 2016

  6. [5]

    H. Gao, C. Xi, K. Li, Q. Zhang, Y. Yang, The Italian Domination Num- ber of Generalized Petersen Graphs P(n,3), Mathematics 2019 7(8), 718, https://doi.org/10.3390/math7080714

    work page doi:10.3390/math7080714 2019

  7. [6]

    H. Gao, T. Xu, Y. Yang, Bagging Approach for Italian Domination inCn2Pm, IEEE Access, PP(99): 1-1, DOI:10.1109/ACESS.2019.2931053

    work page arXiv 2019

  8. [7]

    Hajibaba, N

    M. Hajibaba, N. J. Rad, On Domination, 2-Domination and Italian Domi- nation Numbers, Utilitas Mathematica, 111(2019), 271-280

    work page 2019

Show all 20 references
  1. [8]

    T. W. Haynes, M. A. Henning, L. Volkman, Graphs with Large Italian Domination Number, Bull. Malays. Math. Sci. Soc. (2020), https://doi.org—10.1007—s40840-020-00921-y

    2020

  2. [9]

    T. W. Haynes, M. A. Henning, Perfect Italian Domination in Trees, Discrete Appl. Math., 260 (2019), 164-177

    2019

  3. [10]

    T. W. Haynes, S. T. Hedetniemi, P. J. Slater, Fundamentals of Domination in Graphs, Marcel Dekker, Inc. New York, 1998

    1998

  4. [11]

    M. A. Henning, W. F. Klostermeyer, Italian Domination in Trees , Discrete Appl. Math. 217 (2017), 557-564

    2017

  5. [12]

    A. M. Hinz, S. Klavˇ zar, S. S. Zemljiˇ c, A Survey and Classification of Sierpi´ nski-type Graphs, Discrete Appl. Math., 217 (2017), 565-600

    2017

  6. [13]

    Jismy, S

    V. Jismy, S. Aparna Lakshmanan, Corona Operator on Italian Domination, Communicated

  7. [15]

    Klavˇ zar, U

    S. Klavˇ zar, U. Milutinovi´ c,Graphs S(n,k) and a Variant of the Tower of Hanoi Problem, Czechoslovak Math. J., 47(1) (1997), 95-104. 9

    1997

  8. [16]

    Klavˇ zar, U

    S. Klavˇ zar, U. Milutinovi´ c, C. Petr,1-Perfect Codes in Sierpi´ nski Graphs, Bull. Austral. Math. Soc., 66(3) (2002), 369-384

    2002

  9. [17]

    Lauri, C

    J. Lauri, C. Mitillos, Perfect Italian Domination on Planar and Regular Graphs, arXiv:1905.06293v1 [cs.DM]

    1905 arXiv

  10. [18]

    C. A. Liu, Domination in Sierpi´ nski GraphsS(Kn,t ), arXiv:2008.09807v1 (2020)

    2020 arXiv

  11. [19]

    Poureidi, N

    A. Poureidi, N. J. Rad, On the Algorithmic Complexity of Roman {2}- domination, Iranian Journal of Science and Technology, Transaction A: Sci- ence, 44(6), DOI: 10.1007—s40995-020-00875-7

  12. [20]

    Ramezani, E

    F. Ramezani, E. D. Rodr´ ıguez-Bazan, J. A. Rodr´ ıguez-Vel´ azquez,On the Roman Domination Number of Generalized Sierpi´ nski Graphs, Filomat, 31(20) (2017), 6515-6528. 10

    2017

Pith tools

Reviewed August 27, 2026 · model on record in the stance chip above.