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On the strong geodeticity in the corona type product of graphs

T0 review · 5 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The strong geodetic number of corona-type products is the sum of the strong 2-geodetic numbers of the attached graphs.

desk verdict New strong 2-geodetic parameter and two plausible corona-product formulas, but the edge-corona theorem is undefined as written. read the letter →

arxiv 2411.13139 v1 pith:UXCXOJQI submitted 2024-11-20 math.CO

Bishal Sonar , Satyam Guragain , Ravi Srivastava This is my paper · ORCID
classification math.CO MSC 05C1205C38
keywords stronggeodeticsetnumber2-geodesiccovergeneralizedcoronaproductedgeneighborhood
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 determines the strong geodetic set and strong geodetic number of three corona-type products—the generalized corona, the generalized edge corona, and the generalized neighborhood corona—from the strong 2-geodetic numbers of their component graphs. The central claim is that a strong geodetic basis of the whole product graph is exactly the union of the strong 2-geodetic bases of the attached graphs H_i, so the strong geodetic number is the sum of their strong 2-geodetic numbers. In the edge-corona case, an extra set of base-graph vertices must be added when the base graph has pendent vertices. If correct, the result gives a direct formula for a hard graph parameter on large composite graphs in terms of the smaller factor graphs.

What carries the argument

The central objects are the strong 2-geodesic, a geodesic of length 1 or 2, and the strong 2-geodetic basis, the smallest set of vertices whose assigned unique 2-geodesics cover the whole graph. The mechanism of the proof is locality: in all three corona-type products, each vertex of the base graph G is adjacent (or nearly adjacent, in the neighborhood corona) to vertices from the attached factor graphs, so geodesics of length at most 2 between vertices from different factor bases cover the base vertices, while vertices inside a factor's strong 2-geodetic basis cannot be covered by any geodesic that passes through another factor, forcing the union of the factor bases to be exactly the strong geodetic basis of the product.

What would settle it

Compute the strong geodetic number of the generalized edge corona for a base graph G with |E(G)| ≠ |V(G)|, for example G = P3 (two edges, three vertices) with each factor H_i equal to K2, by brute-force enumeration of all subsets and geodesic covers, and check whether the result equals the sum of the strong 2-geodetic numbers as stated in Theorem 2.12; any mismatch shows the theorem does not hold as stated.

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Extended reading notes

Core claim

The paper establishes that for the generalized corona product G ◦̃ Λ H_i, the generalized edge corona product G ⋄̃ Λ H_i (with an additional term when G has pendent vertices), and the generalized neighborhood corona product G ⋆̃ Λ H_i, the strong geodetic basis is the union of the strong 2-geodetic bases of the factor graphs H_i, and the strong geodetic number equals the sum of the strong 2-geodetic numbers of those factors. The proof runs through three lemmas: each vertex of the base graph G is covered by a fixed geodesic between vertices from two different factor bases, each vertex of a factor graph that lies in its strong 2-geodetic basis cannot be covered by a vertex outside that factor, and the bases are minimal because removing any element breaks coverage. The edge-corona exception arises because a pendent vertex of G can only be covered by a pair from the single factor graph attached to its incident edge, and only when that factor graph has enough distinct length-2 geodesics; otherwise the pendent vertex itself must be added to the strong geodetic set.

Load-bearing premise

The edge-corona theorem is only well-formed when the base graph has as many edges as vertices, since the construction attaches one factor graph per edge but the statement indexes those factor graphs by the vertices of G; the proof also relies on a delicate condition, asserted in Lemma 2.10, about how many length-2 geodesics cover each vertex of a factor graph.

Editorial extensions

If this is right

  • For the generalized corona and generalized neighborhood corona, the strong geodetic number of the product does not depend on the structure of the base graph G beyond its order, since the formula is simply the sum of the strong 2-geodetic numbers of the attached graphs.
  • For the generalized edge corona, the formula holds for base graphs with no pendent vertices, and the only correction for pendent vertices is the addition of the set A of uncovered base vertices, giving a linear dependence on the number of such vertices.
  • When all factor graphs are isomorphic to a single graph H, the results reduce to the ordinary corona, edge corona, and neighborhood corona products, yielding Sg(G ◦ H) = n·Sg′(H), Sg(G ⋄ H) = n·Sg′(H) + |A|, and Sg(G ⋆ H) = n·Sg′(H).
  • Iterating the corona construction, as in the corona graphs G(m+1) = G(m) ◦ G, the strong geodetic number grows as s·n(n+1)^m, following directly from the union formula.

Reading between the lines

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

  • The union-of-bases result suggests that, for any graph operation in which the added parts are attached as leaves or near-leaves to the original graph, the strong geodetic set of the whole will be the disjoint union of the strong 2-geodetic sets of the parts, as long as no part's basis vertex can be covered from outside. That heuristic could be tested on other product families, such as lexicographi
  • The edge-corona exception shows that when a base vertex has degree one, its incident factor graph must supply a pair of basis vertices with a spare length-2 geodesic; this relates the strong geodetic number to the existence of multiple 2-geodesics through each vertex, a condition that could be formalized as a new local parameter of the factor graph.
  • A natural computational test would be to enumerate all connected graphs up to a small order and compare the stated formulas against brute-force strong geodetic numbers, which would quickly expose any hidden structural condition in Lemma 2.10.
  • If the formulas hold, they provide a polynomial-time way to compute the strong geodetic number of corona-type products whenever the strong 2-geodetic numbers of the factors are known, which is notable because the strong geodetic number itself is hard to compute in general.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 4 minor

Summary. The paper studies strong geodetic sets and strong geodetic numbers for three corona-type products: the generalized corona, the generalized edge corona, and the generalized neighborhood corona. For each product, it claims that the strong geodetic basis of the product is the union of strong 2-geodetic bases of the factor graphs H_i, with an additional set A of base-graph vertices in the pendent-vertex case of the edge corona, and that the strong geodetic number is the corresponding sum of strong 2-geodetic numbers. The abstract and the main theorems (Theorems 2.6, 2.12, 2.15) present these reductions as the central contribution.

Significance. If the claimed reductions were correct, they would give a clean method for computing strong geodetic numbers of corona-type products from the easier strong 2-geodetic parameter of the factors. The topic is appropriate for a combinatorics journal, and the idea of relating strong geodeticity to strong 2-geodeticity is worth exploring. However, the manuscript as written contains definitional inconsistencies, a false theorem for the edge-corona product, and an invalid key lemma for the neighborhood-corona case. These are load-bearing failures, not presentation issues. The paper contains no machine-checked proofs or reproducible code, and the small examples do not compensate for the gaps. The contribution cannot be accepted in its present form.

major comments (5)
  1. [Definition 1.2 and Theorem 2.12] Definition 1.2 fixes n=|V(G)|, but the generalized edge corona is defined by joining the two end vertices of the i-th edge of G to every vertex of H_i. This is coherent only when |E(G)|=|V(G)|. For most connected graphs, including P_4, every tree with more than two vertices, and K_{1,3}, the product G ⋄ Λ H_i is not defined. Consequently Theorem 2.12 cannot be instantiated for these graphs, and the edge-corona claim in the abstract is invalid as stated. If the intended definition is n=|E(G)|, then the statement, proof, and Corollary 2.12.1 must be rewritten throughout.
  2. [Theorem 2.12(1)] Even after repairing the indexation, the claimed formula is false. Take G=K2 and H_1=P_3, with the literal edge-corona construction in which both endpoints of the unique edge are joined to every vertex of P_3. In this product graph, the two leaf vertices of P_3 cannot be covered by any geodesic between two other vertices, because all their neighbours are pairwise adjacent; hence any strong geodetic set must contain both leaves. The remaining vertices b, u_1, u_2 cannot all be covered by the single fixed geodesic between the two leaves, since every geodesic between those leaves has length 2 and only one internal vertex. Thus Sg(G ⋄ H_1)=4, while ∑ Sg'(H_i)=Sg'(P_3)=2. This contradicts Theorem 2.12(1) directly.
  3. [Lemma 2.10 and Theorem 2.12(2)] The criterion for covering a pendent base vertex is not established. The proof of Lemma 2.10 asserts that if the 2-geodesic (v_p^a, v_q^a) is needed to cover a vertex u in H_a, then at least three geodesics sharing endpoints v_p^a and v_q^a must exist; but it does not prove that there is no other pair in the strong 2-geodetic basis that could cover u_i, nor that the proposed reassignments can be made simultaneously for all vertices of H_a and all pendent vertices of G. The statement of Theorem 2.12(2) is also unclear: the set A is described through the phrase 'every vertex of that graph is not covered by more than one geodesic of length 2', which is ambiguous as to whether zero coverings are allowed. Because Lemma 2.10 is the only argument for the pendent case, Theorem 2.12(2) is unsupported.
  4. [Lemma 2.13 and Theorem 2.15] Lemma 2.13 is false. Let G=K_3 with vertices u_1,u_2,u_3 and H_i=K_1 for each i; write h_i for the vertex of H_i. For i=2 and j=3, we have u_j=u_3 ∈ N(u_2), as required by the lemma. But the unique geodesic between h_2 and h_3 is h_2-u_1-h_3, which does not cover u_2, and no geodesic between h_2 and h_3 covers u_2. Thus the statement 'u_i can be covered by any geodesic (v_i^p, v_j^q)' fails. Since the proof of Theorem 2.15 relies on Lemma 2.13 to cover every vertex of G, the proof of the neighborhood-corona result is invalid as written. A correct proof would need to choose the covering pair differently for each u_i, and this is not what the lemma states.
  5. [Lower-bound proofs in Theorems 2.6, 2.12, and 2.15] The proofs that the displayed unions are minimal are incomplete. In each theorem, after removing a vertex from ∪ η'_{Sg}(H_i), the proof invokes Lemma 2.5, 2.11, or 2.14 to claim that the removed vertex cannot be covered by one vertex of H_k and one vertex outside H_k. This does not rule out the removed vertex being covered by two vertices both in H_k, or by two vertices both outside H_k. The lemmas concern only cross-factor pairs, so the required exclusion is not supplied. Consequently the lower bound Sg(product) ≥ ∑ s_i is not established even for the generalized corona, where the definitional problem does not arise.
minor comments (4)
  1. [Definitions 2.4 and 2.5] The strong 2-geodetic basis is denoted η'_{Sg}(G) and its cardinality is denoted Sg(G) in Definition 2.5, but the rest of the paper uses Sg'(G) for the strong 2-geodetic number. The notation should be made consistent.
  2. [Corollary 2.15.1] The displayed strong geodetic set uses indices b^1_1,...,b^1_m, where the basis of H has size s, not m. The letter m should be replaced by s.
  3. [Lemma 2.10 proof] The proof contains typographical errors: 'For, v_q^p, v_a^q' should read 'v_p^a, v_q^a', and the phrase 'only a particular vertex (u ∈ V(H_a))' is unclear. These errors make an already vague argument harder to follow.
  4. [Theorem 2.6 proof] The proof refers to 'Lemme 2.5' and 'Lemme 2.11'; the spelling should be 'Lemma'. Also, the proof that removing a vertex destroys the strong 2-geodetic cover of H_i is asserted rather than shown.

Circularity Check

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No circularity found: the strong-geodetic results reduce to the independently defined strong 2-geodetic number of the factor graphs.

full rationale

The paper's central theorems (2.6, 2.12, 2.15) express the strong geodetic number of a corona-type product as a sum of strong 2-geodetic numbers of the factor graphs. The strong 2-geodetic number is defined separately via length-restricted 2-geodesics (Definitions 2.1-2.5) and is not equal by construction to the strong geodetic number of the product graph. The proofs supply explicit covering geodesics through the connecting vertices (Lemmas 2.3, 2.13) and separation arguments (Lemmas 2.5, 2.11, 2.14). No parameter is fitted to the target quantity, no uniqueness theorem from the same authors is invoked, and no ansatz is smuggled in by self-citation; the definitions of the three corona-type products are cited from independent earlier papers. The edge-corona case includes an extra set A for pendent vertices, but A is characterized by the factor graphs' 2-geodesic coverage properties rather than by the desired answer, so it is not a renamed fit. The main weaknesses are correctness issues, not circularity: Definition 1.2 fixes n=|V(G)| while attaching factors to the 'i-th edge', which is ill-defined when |E(G)| differs from |V(G)|, and Lemma 2.10's 'three geodesics' condition is asserted rather than proved. Such defects concern the validity of the theorems, not whether the derivation is equivalent to its inputs.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces the strong 2-geodetic set and number as new graph parameters. These are well-defined mathematical objects and do not constitute 'graviton-style' invented entities. The main unstated assumption is the equality of vertex and edge counts for the edge corona, which is not justified.

assumptions (4)
  • standard math Basic definitions of strong geodetic sets and strong geodetic number from Manuel et al. [11].
    The paper builds on the established definition of strong geodetic sets and numbers.
  • domain assumption The generalized corona, edge corona, and neighborhood corona products are defined as in [5], [1], and [13].
    The product constructions are taken from prior literature, though the edge corona definition in this paper contains an indexing inconsistency.
  • domain assumption Every graph considered is simple and connected with |V(G)| ≥ 2.
    Stated in the preliminaries; the proof of Theorem 2.6 needs n≥2 so that every base vertex lies on a cross geodesic.
  • ad hoc to paper In the generalized edge corona, the number of factor graphs equals the number of edges of G.
    Definition 1.2 says n=|V(G)| but joins the endpoints of the i-th edge to H_i; for the theorem to be well-formed one must assume |E(G)|=|V(G)|, a strong restriction not stated in the paper.

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Pith. "Pith review of On the strong geodeticity in the corona type product of graphs." pith.science (2026). https://pith.science/paper/UXCXOJQI

@misc{pith2026241113139,
  author       = {Pith},
  title        = {Pith review of: On the strong geodeticity in the corona type product of graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UXCXOJQI}},
  note         = {Machine review of arXiv:2411.13139}
}
read the original abstract

The paper focuses on studying strong geodetic sets and numbers in the context of corona-type products of graphs. Our primary focus is on three variations of the corona products: the generalized corona, generalized edge corona, and generalized neighborhood corona products. A strong geodetic set is a minimal subset of vertices that covers all vertices in the graph through unique geodesics connecting pairs from this subset. We obtain the strong geodetic set and number of the corona-type product graph using the strong 2-geodetic set and strong 2-geodetic number of the initial arbitrary graphs. We analyze how the structural properties of these corona products affect the strong geodetic number, providing new insights into geodetic coverage and the relationships between graph compositions. This work contributes to expanding research on the geodetic parameters of product graphs.

Figures

Figures reproduced from arXiv: 2411.13139 by the authors.

Figure 1
Figure 1. Sg(G) = Sg′ (G) = {a, d, e, f} but diam(G) = 3 2.1 Generalized corona product Let G be a graph of order n and H1, H2, . . . , Hn be n graphs of order t1, t2, . . . , tn. We denote the set of vertices by V (G) = {u1, u2, . . . , un}, and V (Hi) = {v i 1 , vi 2 , . . . , vi ti }. The generalized corona is given by G ˜◦ n Λ i=1 Hi , and we have the following: Lemma 2.3. For ui , uj ∈ V (G), then ui , uj can be covered … view at source ↗

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Works this paper leans on

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