{"id":"aef4f063-75a0-435d-9423-be76515bb7fb","arxiv_id":"1909.02718","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A connected bipartite graph has s(G,w)=cs(G,w) for every positive weight function w exactly when it is an even cycle, a double star, a book graph, K3,3 minus an edge, or one of the explicitly listed D or D* graph families.","lead":"This paper determines exactly which connected bipartite graphs always have a cheapest safe set that is connected, for every positive weighting of the vertices. It completes a major case of a problem the same authors posed earlier about when weighted safe numbers and connected weighted safe numbers coincide.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3.6's proof contains an invalid inequality in the |m-n|>=2 case; since the only-if direction of Theorem 1.5 relies on this lemma, the written proof needs a fix.","rationale":"The central classification is well supported: the 'if' direction is backed by explicit constructions and the 'only if' direction is a detailed minimal-counterexample argument. The reader correctly identified Lemma 3.6 as load-bearing, but the specific problem is not merely that Lemma 3.6 is an assumption; the printed proof of Lemma 3.6 contains a concrete algebraic error in the |m-n| >= 2 case. Because Lemma 3.6 yields Corollary 3.8 and is applied repeatedly in Section 5.2, the written proof of Theorem 1.5 is incomplete without a correction to this step. The error is localized and the intended argument can be repaired by using w(S∩Z) > 0, so the theorem itself is likely correct; this is a conditional-accept situation rather than a rejection. I therefore recommend CONDITIONAL acceptance: the authors should fix the proof of Lemma 3.6 and verify that the subsequent applications remain valid unchanged.","tokens_in":27066,"tokens_out":39283,"duration_ms":398606,"concrete_test":"Independently re-derive the second case of Lemma 3.6 using the corrected identity w((X'∪Y)\\S) = alpha(m+n-1) - w(S) + w(S∩Z), together with the fact that the case assumption forces S∩Z ≠ ∅. Verify that 2w(S) >= alpha(m+n-1) + w(S∩Z) contradicts w(S) <= min{m alpha, n alpha} for every m ≠ n, including |m-n| >= 2. If this contradiction holds, the lemma is sound after a proof correction; also audit Corollary 3.8 and Section 5.2 Case 1 to confirm they only use the statement of Lemma 3.6, not the faulty intermediate bound.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"In the proof of Lemma 3.6 (Section 3.2), second case, after Claim 3.7 the paper asserts: w(S) >= w((X'∪Y)\\S) >= w(X'∪Y) - w(S) = alpha(m+n-1) - epsilon(|Z|-1) - w(S). This is algebraically inconsistent: w(X'∪Y) = alpha(m+n-1), so the final equality introduces an unjustified -epsilon(|Z|-1) term. More importantly, w(X'∪Y) - w(S) is not the weight of (X'∪Y)\\S when S∩Z is nonempty; the correct identity is w((X'∪Y)\\S) = w(X'∪Y) - w(S∩(X'∪Y)) = alpha(m+n-1) - w(S) + w(S∩Z). The case assumption (vertices in (X'∪{z}∪Y)\\S are not all in one component) forces S∩Z ≠ ∅, so substituting the correct identity into w(S) >= w((X'∪Y)\\S) gives 2w(S) >= alpha(m+n-1) + w(S∩Z). Since w(S) <= min{m alpha, n alpha} and w(S∩Z) > 0, this contradicts m ≠ n. The paper, however, uses the faulty bound to eliminate |m-n| >= 2 and only treats |m-n| = 1 with the correct S∩Z argument. As written, the exclusionary lemma that powers Corollary 3.8 and the minimal-counterexample proof in Section 5.2 has a gap for |m-n| >= 2. The gap is repairable, but the printed argument is incomplete.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the family Gcs of connected vertex-weighted graphs for which the weighted safe number equals the connected weighted safe number for every positive weight function. The main result, Theorem 1.5, gives a complete classification of connected bipartite graphs in Gcs: even cycles, double stars, book graphs, K_{3,3} minus an edge, and the two families D(m,n;p,q) and D*(m,n;p,q). The proof strategy combines contraction lemmas (Section 3), a chordal-graph characterization (Theorem 1.3), structural analysis of bipartite graphs with dominating edges (Section 4), and a minimal-counterexample argument (Section 5.2).","tokens_in":27372,"tokens_out":6113,"duration_ms":55866,"significance":"If correct, Theorem 1.5 settles the bipartite case of Problem 1.1 raised in [7] and provides a finite, checkable list with a clean structural description. The paper additionally proves Theorem 1.3 for chordal graphs and Theorem 1.4 for triangle-free graphs, and it carefully states its dependence on published results such as [10] and [12]. The contraction-based method is natural for the problem, and the overall proof is organized and detailed. A repaired proof of Lemma 3.6 would make the classification a substantial contribution to the safe-set literature, potentially useful for algorithmic work on connected safe sets.","major_comments":[{"comment":"The displayed inequality w(S) >= w((X'∪Y)\\S) >= w(X'∪Y) - w(S) = alpha(m+n-1) - epsilon(|Z|-1) - w(S) is algebraically incorrect. Since w(X'∪Y) = alpha(m+n-1), the final equality wrongly subtracts epsilon(|Z|-1); the correct identity is w((X'∪Y)\\S) = alpha(m+n-1) - w(S) + w(S∩Z), because S may contain vertices of Z. In the case under consideration, the assumption that (X'∪{z}∪Y)\\S is not contained in one component forces S∩Z ≠ ∅, so substituting the correct identity gives 2w(S) >= alpha(m+n-1) + w(S∩Z) > alpha(m+n-1), which already contradicts w(S) <= alpha min{m,n} for every m ≠ n. Thus the conclusion of Lemma 3.6 is valid, but the printed argument for |m-n| >= 2 is incomplete. Since Lemma 3.6 powers Corollary 3.8 and is applied repeatedly in the minimal-counterexample proof of Theorem 1.5 in Section 5.2, this gap must be repaired in the final version.","section":"Section 3.2, proof of Lemma 3.6, after Claim 3.7"}],"minor_comments":[{"comment":"In the sentence defining a minimum connected weighted safe set, \"w(S) = (G,w)\" should read \"w(S) = cs(G,w)\".","section":"Section 1, definition of cs"},{"comment":"The phrase \"such all pendant vertices in Y\" is grammatically awkward; it should be \"all such pendant vertices in Y\".","section":"Section 5.2, Case |X|=2"}],"recommendation":"major_revision","confidential_remarks":"The error in Lemma 3.6 is genuine and load-bearing for the only-if direction of the main theorem, but it is clearly repairable by the short argument indicated in the report. I therefore recommend major revision rather than acceptance in the current form."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Main take: the paper gives a genuine classification, connected bipartite graphs in Gcs are exactly the listed families, and the proof architecture is credible. The printed proof of Lemma 3.6, which powers the exclusion direction, contains a wrong displayed inequality, but the gap is local and repairable.\n\nWhat is new: Theorem 1.5 settles the bipartite case of Problem 1.1 from [7], and Theorems 1.3 and 1.4 add chordal and triangle-free structural results. The contraction framework (Lemmas 3.1–3.3, 3.6) is a sensible way to turn non-membership into finite obstructions. The central argument is self-contained apart from the cited chordal theorems [10] and [12], which are standard and not the load-bearing risk.\n\nSoft spots: the stress-test note is right about Lemma 3.6. In the second case, after Claim 3.7, the paper claims\nw((X'∪Y)\\S) ≥ w(X'∪Y)−w(S) = α(m+n−1)−ε(|Z|−1)−w(S).\nThe equality is false: w(X'∪Y) = α(m+n−1), with no epsilon term. The correct identity is w((X'∪Y)\\S) = α(m+n−1)−w(S)+w(S∩Z); since the case assumption forces S∩Z ≠ ∅, that identity actually gives a contradiction for every m≠n. So the intended conclusion is true, but the written derivation is not. A referee should ask the authors to rewrite this step; it is a one-line fix, not a structural collapse. Also minor: some subcases are dismissed with \"the same argument\" (e.g., in the proof of Theorem 1.5), which makes verification tedious but not suspicious.\n\nWho this is for: researchers in safe-set problems or graph contractions. They will use Theorem 1.5 directly, especially with the companion FPTAS paper. The paper deserves a serious referee. My recommendation is to send it to peer review and accept after the Lemma 3.6 proof is corrected.","headline":"Real bipartite classification with credible proof architecture, but Lemma 3.6 contains a wrong displayed inequality that needs a one-line fix before acceptance.","tokens_in":27938,"tokens_out":15204,"would_cite":true,"duration_ms":156984,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C69","05C75","05C38"],"pacs":[],"model":"deepseek-v4-flash","headline":"A connected bipartite graph has stable safe sets exactly when it belongs to one of five explicit families of graphs.","keywords":["weighted safe set","connected weighted safe number","bipartite graph","dominating clique","dominating edge","graph contraction","chordal graph","book graph"],"falsifier":"The theorem is false if any listed graph, say $K_{3,3}$ minus an edge, has a positive weight with $s<cs$, or if any unlisted connected bipartite graph has $s=cs$ for every positive weight. A direct check: with all weights equal to 1 on $K_{3,3}$ minus an edge, both numbers should be 3; and for the key lemma, search for a graph contractible to $K_{m,n}$ with $m\\neq n$ and a single connected non-singleton bag that nevertheless satisfies equality on every tested weight.","tokens_in":1821,"feed_emoji":"🔗","tokens_out":2478,"duration_ms":146748,"temperature":0.7,"pith_summary":"This paper determines the complete family of connected bipartite graphs for which the cheapest 'safe' vertex set can always be chosen connected, no matter what positive weights are assigned to the vertices. Here a set is safe if each of its components carries at least as much weight as every adjacent component outside the set. The answer is a short list of five types: even cycles, double stars, book graphs, $K_{3,3}$ with one edge deleted, and two families $D(m,n;p,q)$ and $D^*(m,n;p,q)$ built around a dominating edge. The interest is that the equality $s(G,w)=cs(G,w)$ means the protected region never has to pay an extra capacity premium for being connected; the proof manufactures separating weights from forbidden quotient shapes, then forces any minimal counterexample into the listed forms.","feed_headline":"A complete list of connected bipartite graphs with stable safe sets","feed_subtitle":"Even cycles, double stars, book graphs, and two edge-dominated families are exactly the graphs where connectivity never costs extra.","key_machinery":"The key object is the quotient $\\beta(G,S)$, whose vertices are the components of the chosen set $S$ and of its complement $G-S$, with adjacency when components touch; Lemma 2.1 shows that if $S$ is a minimum safe set witnessing $s<cs$, then $\\beta(G,S)\\notin\\mathcal{G}_{cs}$, so forbidden quotients propagate down to smaller graphs. The main exclusion mechanism is Lemma 3.6: any connected graph contractible to $K_{m,n}$ with $m\\neq n$ and at most one connected bag of size at least two is not in $\\mathcal{G}_{cs}$, because one can weigh the two sides so that every cheap safe set must split across them. Corollary 3.8 turns this into a local obstruction: a high-degree vertex whose neighborhood is independent, whose neighbors all have degree at least two, and whose closed neighborhood does not disconnect the graph forces $G\\notin\\mathcal{G}_{cs}$. The rest of the proof shows that a minimal counterexample would have to be one of the two $D/D^*$ shapes, whose middle layer is either complete bipartite or a double star with a dominating edge, an edge whose endpoints together touch every vertex.","core_discovery":"The central claim is Theorem 1.5: a connected bipartite graph $G$ lies in $\\mathcal{G}_{cs}$ if and only if $G$ is one of (I) an even cycle $C_{2n}$ with $n\\geq 2$; (II) a double star, meaning a tree of diameter at most three; (III) a book graph $B_n$ with $n\\geq 1$; (IV) $K_{3,3}$ with one edge deleted; or (V) a graph $D(m,n;p,q)$ or $D^*(m,n;p,q)$ with $m\\geq 2$, $n\\neq 1$, and $p,q\\geq 0$. Here $\\mathcal{G}_{cs}$ is the family of graphs for which $s(G,w)=cs(G,w)$ for every positive weight function $w$. The 'if' direction verifies each listed family directly; the 'only if' direction takes a minimal counterexample and uses contraction obstructions to force its structure into exactly one of the five forms.","pith_inferences":["Going beyond the paper, a natural testable conjecture is that every graph in $\\mathcal{G}_{cs}$ either contains a dominating clique or is an even cycle; the chordal and bipartite classifications both fit this pattern, and it would reduce the general problem to a finite obstruction search.","Read as a network statement, the theorem says that in any non-listed bipartite network an adversary can choose vertex capacities so that the cheapest safe refuge is fragmented, while a connected refuge requires strictly more capacity; this gives an exact characterization of when connectivity is free.","The quotient-contraction lemmas suggest a recognition strategy for the general family: look for $K_{2,3}$-like quotients with single connected bags. If the same obstruction works beyond bipartite graphs, the problem may have a finite forbidden-quotient characterization."],"forward_implications":["Every connected bipartite graph outside the five families has some positive vertex weights for which $s(G,w)<cs(G,w)$; the paper constructs those weights rather than merely asserting their existence.","Inside the listed families, connectivity is never a liability: for every positive $w$, some connected safe set attains the same minimum weight as the best safe set.","Membership in $\\mathcal{G}_{cs}$ for bipartite graphs becomes decidable by checking a short list of structural conditions; the paper's companion work supplies a linear-time recognition algorithm for that list.","For chordal graphs, Theorem 1.3 gives the same stability exactly when a dominating clique exists, equivalently when the diameter is at most three.","For triangle-free graphs that are not cycles, Theorem 1.4 bounds the diameter by three, so any future search for non-bipartite examples can restrict attention to diameter three."],"supporting_citations":[{"why":"states Problem 1.1 asking for the family Gcs and proves that every cycle satisfies the equality, giving both the motivation and one family in the final list.","marker":"[7]"},{"why":"supplies the theorem that connected chordal graphs contract to a path of length diam(G) with connected bags, used in the proof of Theorem 1.3.","marker":"[10]"},{"why":"supplies the equivalence between having a dominating clique and having diameter at most three in chordal graphs, yielding the double-star characterization for trees.","marker":"[12]"},{"why":"introduces weighted safe sets and the two invariants s(G,w) and cs(G,w) that the whole paper studies.","marker":"[2]"},{"why":"introduces the original safe set problem whose weighted reformulation motivates the equality question.","marker":"[8]"}],"fun_headline_variants":["Even cycles, double stars, books: safe-set equality","Bipartite graphs where connectivity costs nothing","Complete list: graphs with equal safe numbers","The exact bipartite graphs for stable safe sets","When safe numbers don't need connectedness"],"cache_read_input_tokens":29952,"weakest_assumption_plain":"The completeness proof leans on Lemma 3.6, which asserts that any graph contractible to $K_{m,n}$ with $m\\neq n$ and at most one connected bag of size at least two is outside $\\mathcal{G}_{cs}$; if that lemma has an exception, the list of Theorem 1.5 could be missing a graph.","fun_headline_variants_meta":{"raw":{"variants":["Even cycles, double stars, books: safe-set equality","Bipartite graphs where connectivity costs nothing","Complete list: graphs with equal safe numbers","The exact bipartite graphs for stable safe sets","When safe numbers don't need connectedness"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001036,"raw_usage":{"total_tokens":4412,"prompt_tokens":1049,"completion_tokens":3363,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":665,"completion_tokens_details":{"reasoning_tokens":3293}},"tokens_in":665,"tokens_out":3363,"duration_ms":25461,"temperature":1.0,"reasoning_tokens":3293,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T04:42:17.862558+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"The theorem is false if any listed graph, say $K_{3,3}$ minus an edge, has a positive weight with $s<cs$, or if any unlisted connected bipartite graph has $s=cs$ for every positive weight. A direct check: with all weights equal to 1 on $K_{3,3}$ minus an edge, both numbers should be 3; and for the key lemma, search for a graph contractible to $K_{m,n}$ with $m\\neq n$ and a single connected non-singleton bag that nevertheless satisfies equality on every tested weight.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"states Problem 1.1 asking for the family Gcs and proves that every cycle satisfies the equality, giving both the motivation and one family in the final list."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the theorem that connected chordal graphs contract to a path of length diam(G) with connected bags, used in the proof of Theorem 1.3."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the equivalence between having a dominating clique and having diameter at most three in chordal graphs, yielding the double-star characterization for trees."},{"cited_title":"B., Fujita S., Legay S., Manoussakis Y., Matsui Y., Sakuma T","cited_arxiv_id":null,"evidence_quote":"introduces weighted safe sets and the two invariants s(G,w) and cs(G,w) that the whole paper studies."},{"cited_title":"and Sakuma T., Safe set problem on graphs, Discrete Applied Mathematics , 215 (2016), 106–111","cited_arxiv_id":null,"evidence_quote":"introduces the original safe set problem whose weighted reformulation motivates the equality question."}],"review_version":1}