{"id":"66523038-d33d-4343-b3b2-9272371523bf","arxiv_id":"2411.13139","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The strong geodetic number of generalized corona and neighborhood corona products equals the sum of the factor graphs' strong 2-geodetic numbers, while the edge corona version adds corrections for pendent vertices.","lead":"This paper proposes formulas for the strong geodetic number of three corona-type graph products in terms of a new parameter, the strong 2-geodetic number of the factor graphs. A smart generalist might read it to see how gluing graphs together affects the size of the smallest set of vertices that covers every vertex with unique shortest paths.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Definition 1.2 fixes n=|V(G)| but assigns H_i to the i-th edge, so Theorem 2.12 is undefined whenever |E(G)| differs from |V(G)|.","rationale":"The reader identified the same mismatch (Definition 1.2 sets n=|V(G)| while one factor graph is assigned to each edge) as the weakest point. My independent reading of the text confirms that there is no way to make Theorem 2.12 well-formed for graphs such as P_4: indexing edges by the n vertices only works when |E(G)|=|V(G)|. Since the abstract and the paper's main claim cover the generalized edge corona as one of three central products, an ill-defined central theorem is sufficient to reject the paper as it stands. I do not ground the rejection in disagreement with the intended formula; the generalized corona and neighborhood corona reductions are plausible and might be salvageable, but the edge-corona theorem as written is uninstantiable for most inputs. The proposed P_4 check is immediate and would settle the structural concern.","tokens_in":10927,"tokens_out":23503,"duration_ms":244113,"concrete_test":"Check the well-formedness of Definition 1.2 on G=P_4 with vertices u_1-u_2-u_3-u_4 and edges e_1=u_1u_2, e_2=u_2u_3, e_3=u_3u_4. Since n=|V(G)|=4 but |E(G)|=3, there is no fourth edge e_4; any choice of H_1,H_2,H_3,H_4 leaves H_4 unattached, so the generalized edge corona and Theorem 2.12 are undefined for this input. This settles the concern because the paper's statement is meant to cover arbitrary connected G, and P_4 is such a graph.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Definition 1.2 and Theorem 2.12 are inconsistent as stated. Definition 1.2 fixes n=|V(G)| and constructs the generalized edge corona by 'joining the two end vertices of ith edge of G to every vertex of H_i', so it silently assumes that the edges of G are indexed by the n vertices. For any connected graph with |E(G)| ≠ |V(G)| — for example P_4, every tree with n>2, or K_{1,3} — one of H_1,...,H_n has no edge to be attached to, so the generalized edge corona is not defined. Lemma 2.10 and Theorem 2.12 are stated with the same |V(G)|=n hypothesis, so the claimed formula Sg(G⋄Λ H_i)=Σ s_i (+|A|) cannot even be instantiated for most graphs. This invalidates one of the three central products claimed in the abstract. Even if the intended correction is n=|E(G)|, the paper would still need a rigorous treatment of the pendent-vertex case: Lemma 2.10's 'three geodesics' criterion is asserted rather than derived, and the proof does not show that the fixed 2-geodesics used to cover H_i can be chosen simultaneously with those covering u_i.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":11073,"tokens_out":25968,"duration_ms":240489,"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":[{"comment":"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.","section":"Definition 1.2 and Theorem 2.12"},{"comment":"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.","section":"Theorem 2.12(1)"},{"comment":"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.","section":"Lemma 2.10 and Theorem 2.12(2)"},{"comment":"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.","section":"Lemma 2.13 and Theorem 2.15"},{"comment":"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.","section":"Lower-bound proofs in Theorems 2.6, 2.12, and 2.15"}],"minor_comments":[{"comment":"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.","section":"Definitions 2.4 and 2.5"},{"comment":"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.","section":"Corollary 2.15.1"},{"comment":"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.","section":"Lemma 2.10 proof"},{"comment":"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.","section":"Theorem 2.6 proof"}],"recommendation":"reject","confidential_remarks":"The reader's report identified the Definition 1.2 / |E(G)| versus |V(G)| inconsistency correctly. My independent check shows the problem is worse than undefinedness: the edge-corona theorem is false for G=K_2 and H_1=P_3 once the definition is interpreted literally. The false Lemma 2.13 for the neighborhood corona is also a substantive error, not a stylistic gap. These issues cannot be repaired by local edits within the scope of the current manuscript, since the correct statements and proofs would require a substantially different analysis."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe short version: the paper introduces a genuinely new parameter and has two plausible theorems, but the edge-corona section is broken by a definitional mismatch. Definition 1.2 fixes n=|V(G)| yet assigns each H_i to the \"i-th edge\" of G, so the generalized edge corona product is undefined whenever |E(G)| ≠ |V(G)|. Theorem 2.12 leans on that product, so one of the three headline results doesn't hold as stated.\n\nWhat's real here: the strong 2-geodetic number is a reasonable variant of the strong geodetic number, and the reductions for the generalized corona and generalized neighborhood corona — Sg of the product equals the sum of the strong 2-geodetic numbers of the factors — look correct in structure. I checked the broad argument: the union of the strong 2-geodetic bases of the H_i's strongly geodetically covers the product, and Lemma 2.5 (vertices in a factor's basis can't be covered from outside the factor) is essentially right, though the proof is terse. Lemma 2.4 about distances collapsing to 2 is fine.\n\nThe soft spots: the edge-corona problem is not a minor typo, because the rest of that section (Lemma 2.10, Theorem 2.12, Corollary 2.12.1) inherits it. Even if you change n to |E(G)|, Lemma 2.10's \"three geodesics\" condition is asserted rather than derived; I can't tell whether the simultaneous geodesic-fixing argument works. Also, the proof of Theorem 2.6 doesn't explicitly show that every vertex of G is covered by some cross-pair geodesic; I think it's true because each basis vertex pairs with a vertex from another factor and covers its attached G-vertex as an endpoint, but the paper should say that. The notation is careless in a few places (\"N+1 graphs\").\n\nNet: I'd send this to a referee rather than desk-reject. The strong 2-geodetic notion is worth airing, and the corona and neighborhood-corona results are probably salvageable. The edge-corona part needs to be rebuilt from a corrected definition. If the authors fix that, it could be a decent paper.\n\nFor your question: I'd give this a \"maybe\" for reading group — useful as a case study in how a single indexing error can sink a claimed generalization. I wouldn't cite it yet in its current form.","headline":"New strong 2-geodetic parameter and two plausible corona-product formulas, but the edge-corona theorem is undefined as written.","tokens_in":11709,"tokens_out":7375,"would_cite":false,"duration_ms":67718,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C12","05C38"],"pacs":[],"model":"deepseek-v4-flash","headline":"The strong geodetic number of corona-type products is the sum of the strong 2-geodetic numbers of the attached graphs.","keywords":["strong geodetic set","strong geodetic number","2-geodesic","strong 2-geodesic cover","generalized corona product","generalized edge corona product","generalized neighborhood corona product","geodetic number"],"falsifier":"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.","tokens_in":10607,"feed_emoji":"📐","tokens_out":2853,"duration_ms":39776,"temperature":0.7,"pith_summary":"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.","feed_headline":"Corona graphs' strong geodetic number is a sum of factor parts","feed_subtitle":"New formulas compute the strong geodetic number of three corona-type products directly from the attached graphs' small-scale parameters.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Introduces the strong geodetic set and strong geodetic number, the central parameters studied in the paper.","marker":"[11]"},{"why":"Defines the generalized corona product, the first of the three product operations whose strong geodetic number is computed.","marker":"[8]"},{"why":"Defines the generalized edge corona product, the setting for the main exception where pendent vertices add to the strong geodetic set.","marker":"[1]"},{"why":"Defines the generalized neighborhood corona product, the third product treated by the paper.","marker":"[13]"},{"why":"Provides the iterative corona graph construction whose strong geodetic number is derived as a corollary of the corona product theorem.","marker":"[14]"}],"fun_headline_variants":["Corona product geodetic number equals sum of factor 2-geodetics","Strong geodetic sets in corona graphs built from factor bases","Edge corona adds pendent vertex term to geodetic sum","Generalized corona products: geodetic number sums from factors"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Corona product geodetic number equals sum of factor 2-geodetics","Strong geodetic sets in corona graphs built from factor bases","Edge corona adds pendent vertex term to geodetic sum","Generalized corona products: geodetic number sums from factors"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00025,"raw_usage":{"total_tokens":1539,"prompt_tokens":916,"completion_tokens":623,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":532,"completion_tokens_details":{"reasoning_tokens":548}},"tokens_in":532,"tokens_out":623,"duration_ms":43581,"temperature":1.0,"reasoning_tokens":548,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T16:50:02.532969+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Manuel, S","cited_arxiv_id":null,"evidence_quote":"Introduces the strong geodetic set and strong geodetic number, the central parameters studied in the paper."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the generalized corona product, the first of the three product operations whose strong geodetic number is computed."},{"cited_title":"On Generalized Edge Corona Product of Graphs","cited_arxiv_id":"1712.04699","evidence_quote":"Defines the generalized edge corona product, the setting for the main exception where pendent vertices add to the strong geodetic set."},{"cited_title":"Setiawan et al","cited_arxiv_id":null,"evidence_quote":"Defines the generalized neighborhood corona product, the third product treated by the paper."},{"cited_title":"Sharma, B","cited_arxiv_id":null,"evidence_quote":"Provides the iterative corona graph construction whose strong geodetic number is derived as a corollary of the corona product theorem."}],"review_version":1}