{"id":"841692a5-ed6a-4ec5-81cc-cc8b1cf46f8e","arxiv_id":"1908.06859","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"Exact formulas and bounds are claimed for double Roman domination under Cartesian products, coronas, and twin addition, with several lower-bound proofs resting on invalid assumptions.","lead":"This paper studies how the double Roman domination number of a graph changes under Cartesian products, corona products, and adding twin vertices. It reports exact formulas and bounds, but several key proofs contain unstated or false assumptions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2.5's lower bound assumes closed neighborhoods of a dominating set partition G_{2,n}; for n=3 this is false, so the exact grid formula is unsupported.","rationale":"The reader's weakest-assumption identification is exactly the load-bearing failure: Theorem 2.5 derives the headline exact value from a partition property of closed neighborhoods of dominating sets that is false for n=3. This is not a matter of consensus or style; the proof text states the false assertion directly, and no alternative lower-bound argument appears in the paper. Secondary but reinforcing, Proposition 3.3 constructs a function with f(u_i)=2 while the unique leaf neighbor u'_i is assigned 0, leaving that leaf with only one neighbor of value 2, so the constructed function is not a valid double Roman dominating function. This additional flaw supports rejection but does not change the verdict, since the grid proof is already invalid as written. The proposed exact computation would determine whether the grid formula itself is true; if it is, the paper would still need new proofs before the claim could be accepted.","tokens_in":10749,"tokens_out":17079,"duration_ms":189307,"concrete_test":"Compute gamma_dR(G_{2,n}) exactly for n = 1 and n = 3,...,10 (n=2 is excluded by the theorem) by exhaustive search over assignments in {0,2,3} or with an integer linear program, and compare each value with floor((3n+4)/2). If any value differs, the exact claim is false. If all values match, the formula survives but the paper still lacks a valid lower-bound proof, so a replacement argument would be required before acceptance.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's main exact result for Cartesian products is gamma_dR(G_{2,n}) = floor((3n+4)/2), proved in Theorem 2.5. The only lower-bound argument for odd n asserts that for any dominating set {x_1,...,x_gamma} of G_{2,n}, the closed neighborhoods N[x_i] partition the vertex set and each has size > 3. This is contradicted by n=3: the two middle vertices form a minimum dominating set, and their closed neighborhoods overlap (each contains the other), with sizes 4 and 4, so they cannot partition the six vertices. For even n the proof similarly assumes an arbitrary dominating set can be refined to a partition A_i with A_i subset of N[x_i] and with at most one singleton block; no justification is supplied. Since no alternative lower-bound proof is given, the claimed equality for all n != 2 rests on this false premise. The upper bound may be correct, but equality is not established by the submitted argument.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies the double Roman domination number γ_dR(G) under graph operations. It presents a lower bound for Cartesian products (Theorem 2.2), an upper bound (Theorem 2.4), an exact formula for the 2×n grid (Theorem 2.5), exact values and realizability results for corona products G⊙H (Section 3), bounds and exact values for adding true/false twins (Section 4), and a realizability result for pairs (γ_R, γ_dR) on connected bipartite graphs (Section 5). The main claimed advances are the grid formula and the corona values, which would extend the known theory of double Roman domination to new graph families.","tokens_in":11011,"tokens_out":15614,"duration_ms":140576,"significance":"If the results were correctly proved, they would give the first exact double Roman domination numbers for 2×n grids and several corona families, and would complement known bounds for Roman domination. The paper contains explicit and easily checked upper-bound constructions, and the general upper bound in Theorem 2.4 is a useful observation. However, the lower-bound proofs for the main exact results are not valid as written: several rest on false or unjustified assertions, including the closed-neighborhood partition in Theorem 2.5, the lower-bound claim in Proposition 3.1, and the construction in Proposition 3.3. Since these arguments are load-bearing for the central claims, the current manuscript does not establish its results.","major_comments":[{"comment":"The lower-bound proof for odd n asserts that for any dominating set {x_1,...,x_γ} of G_{2,n}, the closed neighborhoods N[x_i] form a partition of the vertex set. This is false: for n=3, the minimum dominating set {(u1,v2),(u2,v2)} has N[(u1,v2)] = {(u1,v1),(u1,v2),(u1,v3),(u2,v2)} and N[(u2,v2)] = {(u2,v1),(u2,v2),(u2,v3),(u1,v2)}, which overlap at two vertices. The claimed lower bound γ_dR(G_{2,n}) ≥ 3(n+1)/2 depends on this partition assertion, so the inequality is not established.","section":"§2, Theorem 2.5"},{"comment":"The proof asserts without justification that the set S_v = (S - {u_i : Z_v^i=1}) ∪ X_v is a dominating set for G. This is false for arbitrary dominating sets: for G=P4 with vertices 1-2-3-4 and S={2,3}, if one removes 2 and X_v={3}, then S_v={3} does not dominate vertex 1. No property of the specific f is supplied that rules out such a configuration. In addition, the chain deriving a strict inequality uses '>' at a point where the preceding display only supports '≥'; hence the strict form of Theorem 2.2 is not proved.","section":"§2, Theorem 2.2"},{"comment":"The lower-bound argument states that 'there are n mutually exclusive copies of H each of which requires at least weight 3 in a DRDF'. This is not true in the form stated: if the attachment vertex u_i in G has value 3, all vertices of the copy H_i may receive value 0, so that copy contributes no positive weight. The total lower bound 3n may still be true, but the argument given does not prove it.","section":"§3, Proposition 3.1"},{"comment":"The constructed function f is not a DRDF. For i=2,...,m+1, the leaf vertex u'_i is assigned 0 while its only neighbor u_i is assigned 2. A vertex with value 0 requires either two neighbors in V2 or one neighbor in V3, so u'_i is not double Roman dominated. Thus the claimed realization of weight 3n-m is invalid, and the statement that every value between 2n+1 and 3n is realizable is not demonstrated.","section":"§3, Proposition 3.3"},{"comment":"The proofs of these exact values are incomplete. Proposition 3.4 asserts the existence of a γ_dR-function of P_n with V2=∅ for n=3k or 3k+2 without construction or proof, and the transformation g is not shown to be a minimum DRDF. Proposition 3.5 says only that the proof is 'similar to that of P_n' and supplies no details. Since these are claimed exact values, the lower bounds are missing.","section":"§3, Propositions 3.4 and 3.5"},{"comment":"The reverse-inequality case analysis contains an error. For k=p+q-1, the displayed inequality g(V)=2(p+q)+1 > 2k+3(p+q-k) is actually an equality, not a strict inequality, so the contradiction does not follow. Consequently the lower bound γ_dR(K_{p,q}⊙K1) ≥ 2(p+q+1) for p,q≥2 is not proved. Also, the case split says 'p,q>2' where the statement covers p,q≥2.","section":"§3, Proposition 3.7"}],"minor_comments":[{"comment":"The explicit DRDF formula is not correctly specified: with the printed condition 'k>0' for the i=2 row, the vertex (u1,v1) has no positive neighbor for n=3 and n=5, so the stated f is not a DRDF. If the intended condition is k≥0, the formula and its verification should be corrected.","section":"§2, Theorem 2.5"},{"comment":"There are typos: 'vetrtex' should be 'vertex', and '1−1 correspondence' should be 'one-to-one correspondence'.","section":"§1, Introduction"},{"comment":"The assertion that the case f(u)=2 and f(u')=3 in a γ_dR-function 'won't arise' needs justification; if it can arise, a swap of the values of u and u' would handle it, but this should be stated.","section":"§4, Theorem 4.1"},{"comment":"Figure 1 is referenced in Theorem 2.5 but is not displayed in the manuscript text.","section":"Figure 1"}],"recommendation":"reject","confidential_remarks":"The issues in Theorem 2.5 and Section 3 are not isolated typos; the lower-bound arguments are structurally unsound, and several exact-value claims rest on assertions that are either false or unproved. In my view, the manuscript would need substantially new proofs of the central claims, which goes beyond a routine revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here is my read of arXiv:1908.06859. The paper extends the double Roman domination program to Cartesian products, coronas, and twin addition. The questions are natural, and the paper catalogs plausible exact values: gamma_dR(G_{2,n}) = floor((3n+4)/2) for n != 2, gamma_dR(G ⊙ H) = 3n for H not K1, bounds for G ⊙ K1, and twin-addition bounds. Some of these are genuinely new relative to the cited literature, and the easy upper-bound constructions are clean. Proposition 3.1, Theorem 2.4, and the general twin bounds are fine in spirit.\n\nThe soft spots are load-bearing. Theorem 2.5's lower bound asserts that for odd n the closed neighborhoods of an arbitrary dominating set partition the vertex set of G_{2,n}. That is false: for n = 3, the two middle vertices form a dominating set whose closed neighborhoods overlap. The even case's partition with at most one singleton block is also asserted without proof. The claimed equality therefore rests on an invalid partition argument; the upper bound alone does not establish the result.\n\nProposition 3.3's converse construction also fails. In the corona of K_{1,m} ∪ (n-m-1)K_1 with K_1, the leaf u'_j for j = 2,...,m+1 is assigned 0 while its support u_j gets 2. A zero vertex needs two neighbors in V2 or one in V3; that leaf has exactly one neighbor total, so the condition fails. The weight calculation is correct, but it computes the weight of a function that is not a DRDF.\n\nThere are smaller gaps too. Theorem 4.1's left-inequality proof sketches cases without fully justifying why a gamma_dR-function on H restricts to one on G; the true-twin case needs a minimality argument that is only gestured at. Theorem 5.2's \"easy to verify that these functions are in fact minimum\" is a promise, not a proof.\n\nThe core issue is not the topic or the results — the formulas may well be true. But as written the proofs do not support the claims, and a referee would need substantial new arguments for at least two of the main results. A serious referee could still find value in the upper bounds and the clean parts, but this is not ready for publication. I would reject in current form or require major revision.","headline":"New formulas for a natural variant of Roman domination, but two central proofs do not hold as written: the exact grid value and the corona realization are unsupported.","tokens_in":11354,"tokens_out":6873,"would_cite":false,"duration_ms":72795,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C69","05C76"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper determines the double Roman domination number exactly for $2\\times n$ grids and for corona products, and bounds it under Cartesian products and twin additions.","keywords":["double Roman domination","Roman domination","domination number","Cartesian product","grid graph","corona product","true and false twins","realizability of domination parameters"],"falsifier":"Enumerate all $3^6$ assignments $V\\to\\{0,2,3\\}$ on the $2\\times3$ grid and check the two double Roman conditions: the formula predicts $\\gamma_{dR}=6$, so an assignment of weight 5 would refute Theorem 2.5, while confirming 6 would leave the formula intact for that case but still show the partition assertion used in the proof is false.","tokens_in":10565,"feed_emoji":"🕸️","tokens_out":19087,"duration_ms":171332,"temperature":0.7,"pith_summary":"Double Roman domination is a graph-protection cost model: each vertex receives 0, 2, or 3, a vertex receiving 0 must see either two neighbors receiving 2 or one neighbor receiving 3, and in a minimum solution no vertex need receive 1. The paper works out how the minimum total weight behaves when graphs are combined by Cartesian products, corona products (attaching a copy of one graph to each vertex of another), and twin additions. It gives the exact value for the $2\\times n$ grid as $\\lfloor(3n+4)/2\\rfloor$, and shows that the corona $G\\odot H$ with any nontrivial $H$ costs exactly $3|V(G)|$, while $G\\odot K_1$ costs between $2|V(G)|+1$ and $3|V(G)|$, with every integer in that interval realized. For Cartesian products it proves $\\gamma_{dR}(G\\square H)>\\gamma(G)\\gamma_{dR}(H)/2$ together with an upper bound in terms of the order of either factor. It also rules out the pair $(\\gamma_R,\\gamma_{dR})=(a,a+1)$ for even $a>2$ and realizes a broad range of other pairs on connected bipartite graphs.","feed_headline":"Double Roman number of 2-by-n grids is floor((3n+4)/2)","feed_subtitle":"Exact values also for corona products; Cartesian products and twin additions get explicit bounds.","key_machinery":"The carrying object is the double Roman dominating function (DRDF), a map $f:V\\to\\{0,2,3\\}$ whose weight $f(V)$ is minimized, together with the fact that a minimum DRDF need never use the value 1. For Cartesian products the main mechanism is a slice-and-project argument: take a minimum dominating set of one factor, partition that factor into closed-neighborhood pieces, slice the product over those pieces, push the inherited assignment down to the other factor, and compare weights; binary vectors record which slices fail, and a domination argument on the other factor bounds their total. For the grid, the upper bound comes from an explicit periodic placement of 3s with one 2 on the end for even $n$, while the lower-bound argument assigns weight to the members of a minimum dominating set and uses the closed neighborhoods around them. For coronas and twins, the machinery is local forcing: pendant vertices must be protected from outside their own edge, and twin vertices can be handled by restricting or re-assigning a minimum function.","core_discovery":"The central claim is that several common graph operations change the double Roman domination number in a controlled, mostly computable way. On the $2\\times n$ grid $G_{2,n}$ with $n\\neq2$, the paper asserts $\\gamma_{dR}(G_{2,n})=\\lfloor(3n+4)/2\\rfloor$, achieved by a periodic pattern of 3s and, for even $n$, a single 2 at the far end. For the corona $G\\odot H$ with $H\\not\\cong K_1$, it claims $\\gamma_{dR}=3|V(G)|$; for $G\\odot K_1$, the value lies in the interval $[2|V(G)|+1,\\,3|V(G)|]$ and every integer in that interval occurs. For a Cartesian product, the paper proves $\\gamma_{dR}(G\\square H)>\\gamma(G)\\gamma_{dR}(H)/2$ and $\\gamma_{dR}(G\\square H)\\le\\min\\{|V(H)|\\gamma_{dR}(G),|V(G)|\\gamma_{dR}(H)\\}$. Adding a true twin raises the number by at most 1, and a false twin by at most 2, and neither addition can lower it. Finally, no graph realizes $(\\gamma_R,\\gamma_{dR})=(a,a+1)$ when $a>2$ is even, whereas every pair $(a,b)$ with $a=\\lfloor b/2\\rfloor+1,\\ldots,2\\lfloor b/2\\rfloor-1$ is realized by some connected bipartite graph.","pith_inferences":["The periodic 3/2 construction in the grid theorem gives a ready upper bound for $2\\times n$ grids; if a correct lower-bound argument replaces the closed-neighborhood partition step, the same pattern is a natural starting point for exact values on wider $m\\times n$ grids.","The slice-and-project argument suggests a Vizing-type product inequality: $\\gamma_{dR}(G\\square H)$ should be bounded below by a constant factor of $\\gamma(G)\\gamma_{dR}(H)$, and similar ratio bounds may hold for total Roman and other Roman-type parameters.","The constructions that tune $G\\odot K_1$ and the bipartite realization graphs give a way to force $\\gamma_{dR}$ to any prescribed value in an interval while keeping the graph structurally simple, which may serve as a building block for hardness or approximation results.","The exclusion of $(a,a+1)$ for even $a>2$ leaves open which pairs close to $\\gamma_{dR}=2\\gamma_R$ are realizable; the paper's range covers most of the interval, and the remaining gap is a concrete finite search problem."],"forward_implications":["For any graph $G$ and any nontrivial $H$, the corona $G\\odot H$ has $\\gamma_{dR}=3|V(G)|$, so the double Roman number of such a corona is determined entirely by the size of the base graph.","For $G\\odot K_1$, the parameter can be tuned to every integer between $2|V(G)|+1$ and $3|V(G)|$, and for paths and cycles the paper gives explicit values asymptotic to $7n/3$.","The product inequalities place $\\gamma_{dR}(G\\square H)$ strictly above $\\gamma(G)\\gamma_{dR}(H)/2$ and at most $\\min\\{|V(H)|\\gamma_{dR}(G),|V(G)|\\gamma_{dR}(H)\\}$, so both factors constrain the product's cost.","Adding a true twin increases the double Roman number by at most 1 and adding a false twin by at most 2, and neither operation ever lowers it.","The realizability results show which pairs $(\\gamma_R,\\gamma_{dR})$ coexist, excluding $(a,a+1)$ for even $a>2$ while covering every value between $\\lfloor b/2\\rfloor+1$ and $2\\lfloor b/2\\rfloor-1$ on connected bipartite graphs."],"supporting_citations":[{"why":"Introduces the double Roman dominating function, the no-1-in-minima fact, and the bounds $2\\gamma(G)\\le\\gamma_{dR}(G)\\le3\\gamma(G)$ and $\\gamma_R(G)<\\gamma_{dR}(G)<2\\gamma_R(G)$ that frame every result.","marker":"[5]"},{"why":"Supplies the cycle formula $\\gamma_{dR}(C_n)$ used for the base case $G_{2,2}\\cong C_4$ and for comparison in the realizability discussion.","marker":"[1]"},{"why":"Gives the lower bound $\\gamma_R(G\\square H)>\\gamma(G)\\gamma(H)$ used to derive the opening relationship $\\gamma_{dR}(G\\square H)>\\gamma(G)\\gamma(H)$ for Cartesian products.","marker":"[11]"},{"why":"One of the sources for the realized pair $(a,a+\\lceil a/2\\rceil)$ on paths, needed to delimit the cases considered in the realizability theorems.","marker":"[3]"},{"why":"Another source for the path realization $(a,a+\\lceil a/2\\rceil)$ cited in Section 5 before the new realizability constructions.","marker":"[10]"}],"fun_headline_variants":["Exact double Roman numbers for 2×n grids and corona products","Cartesian product bounds for double Roman domination","Double Roman number: exact on grids, bounded on twins","Twin addition: double Roman number changes by at most 2"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that in the lower-bound proof for the $2\\times n$ grid, the closed neighborhoods $N[x_i]$ of a dominating set $\\{x_1,\\ldots,x_\\gamma\\}$ partition the vertex set; this is false for $n=3$ with the dominating set $\\{a_2,b_2\\}$, whose closed neighborhoods overlap, so the proof of Theorem 2.5 is not supported as written.","fun_headline_variants_meta":{"raw":{"variants":["Exact double Roman numbers for 2×n grids and corona products","Cartesian product bounds for double Roman domination","Double Roman number: exact on grids, bounded on twins","Twin addition: double Roman number changes by at most 2"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001484,"raw_usage":{"total_tokens":6021,"prompt_tokens":1066,"completion_tokens":4955,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":682,"completion_tokens_details":{"reasoning_tokens":4886}},"tokens_in":682,"tokens_out":4955,"duration_ms":42352,"temperature":1.0,"reasoning_tokens":4886,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:34:05.341070+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Enumerate all $3^6$ assignments $V\\to\\{0,2,3\\}$ on the $2\\times3$ grid and check the two double Roman conditions: the formula predicts $\\gamma_{dR}=6$, so an assignment of weight 5 would refute Theorem 2.5, while confirming 6 would leave the formula intact for that case but still show the partition assertion used in the proof is false.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the double Roman dominating function, the no-1-in-minima fact, and the bounds $2\\gamma(G)\\le\\gamma_{dR}(G)\\le3\\gamma(G)$ and $\\gamma_R(G)<\\gamma_{dR}(G)<2\\gamma_R(G)$ that frame every result."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the cycle formula $\\gamma_{dR}(C_n)$ used for the base case $G_{2,2}\\cong C_4$ and for comparison in the realizability discussion."},{"cited_title":"Wu, An Improvement on Vizing’s Conjecture , Inform","cited_arxiv_id":null,"evidence_quote":"Gives the lower bound $\\gamma_R(G\\square H)>\\gamma(G)\\gamma(H)$ used to derive the opening relationship $\\gamma_{dR}(G\\square H)>\\gamma(G)\\gamma(H)$ for Cartesian products."},{"cited_title":"Math., 244 (2018), 198-204","cited_arxiv_id":null,"evidence_quote":"One of the sources for the realized pair $(a,a+\\lceil a/2\\rceil)$ on paths, needed to delimit the cases considered in the realizability theorems."},{"cited_title":"Xu, Roman Domination , http://www.math.uchicago","cited_arxiv_id":null,"evidence_quote":"Another source for the path realization $(a,a+\\lceil a/2\\rceil)$ cited in Section 5 before the new realizability constructions."}],"review_version":1}