{"id":"dc21d51e-a15a-4628-8f09-a1d963eae5bc","arxiv_id":"2412.03869","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Every connected graph with n≥7 vertices and at most floor(3n/2) edges has an independent minimum vertex cut; every connected graph with n≥7 and at most 2n edges has a foresty minimum vertex cut; both bounds are sharp.","lead":"Two theorems give tight edge thresholds for sparse connected graphs to contain an independent minimum vertex cut and to contain a foresty minimum vertex cut. Both thresholds, floor(3n/2) and 2n edges, are shown to be best possible.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Base-case checks in the proofs of Theorems 3 and 4 are asserted rather than demonstrated; if any listed n=7 case is incomplete or incorrect, both inductions collapse.","rationale":"The reader's weakest assumption is exactly the one I would choose: the n=7 base cases are asserted rather than proved. I checked the surrounding machinery, including Lemma 6, Lemma 9, and the induction steps for both theorems, and found those arguments coherent. The induction hypotheses apply only from order 8 onward, so the whole proof depends on a complete and correct finite enumeration at n=7. This is load-bearing, but it is also mechanical and independently checkable; I found no actual counterexample among the cases I considered. Therefore the concern does not overturn the reader's ACCEPT verdict, but it justifies asking for an explicit verification of the listed base cases before relying on the proof as written.","tokens_in":6216,"tokens_out":36788,"duration_ms":374543,"concrete_test":"Independently enumerate all connected graphs of order 7 with size at most 10 (Theorem 3) and size at most 14 (Theorem 4), or equivalently all four-vertex graphs H=F−N[v] compatible with Lemma 5 and the size/degree bounds, and mechanically verify that every graph in the listed cases has the stated independent or foresty minimum cut. If every case passes, the concern is resolved; if any case fails, the corresponding theorem's base case fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Theorem 3's basis step, after reducing to κ=2 and a degree-2 vertex v with N(v)={x,y}, xy∈E, the authors list nine possibilities for H=F−N[v] and state that in each case “it is easy to find two nonadjacent vertices which form a vertex cut.” The enumeration itself is not derived: one must impose Lemma 5, the degree-2 condition, and e(F)≤10, and the printed list depends on notation (K1∨K3 is K4 if K3 is a triangle, but would be K1,3 only if an overline on K3 was lost in typesetting; these are different four-vertex graphs and only one can actually occur). No individual case is verified. In Theorem 4's basis step, when δ=3 and N(v)=C3, the cases M[R]∈{K2+K1,P3,C3} are disposed of by “easy to verify” and an unstated appeal to Lemma 8, again without exhibiting the promised foresty 3-cut. These finite checks are the load-bearing foundation of both inductions: a single omitted case, or a case in which the claimed cut is not a cut, would invalidate Theorems 3 and 4. I found no independent flaw in Lemma 6, Lemma 9, or the induction steps, so this is the weakest point.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves two extremal results about minimum vertex cuts in sparse connected graphs. Theorem 3 states that every connected graph of order n at least 7 with at most floor(3n/2) edges has an independent minimum vertex cut, and that the edge bound is best possible. Theorem 4 states the analogous result with bound 2n for foresty minimum vertex cuts. The proofs are by induction on n, with auxiliary lemmas for cubic and 4-regular graphs, together with explicit sharpness constructions. The paper also recalls and uses standard facts such as Lemma 5 and a Menger-type matching lemma (Lemma 8).","tokens_in":6484,"tokens_out":12876,"duration_ms":140211,"significance":"If the results are correct, they give clean exact edge thresholds for the existence of structured minimum vertex cuts, extending the earlier Chen-Yu and Le-Pfender results on arbitrary independent cuts to minimum cuts, and contributing the new notion of foresty minimum cuts in the sparse regime. The proof strategy is self-contained and elementary, with no fitted parameters or reliance on prior results of the authors. The main gap is the finite verification of the n=7 induction bases, which is asserted rather than written out; once supplied, I see no obstacle to the main theorems. The sharpness constructions are explicit and have the claimed edge counts, which is a genuine strength.","major_comments":[{"comment":"The basis step is asserted rather than demonstrated. The derivation of the list F−N[v] ∈ {2K1+K2, 2K2, K1+P3, K1+C3, K1∨K3, P4, C4, K1∨(K1+K2), K4−e} is not given, and the sentence 'In each case it is easy to find two nonadjacent vertices which form a vertex cut by using the size condition' suppresses the actual case analysis. This is the entire base of the induction, so a missing case or an incorrect cut in any listed case would invalidate Theorem 3. In addition, the entry 'K1∨K3' is ambiguous: if K3 is the triangle, the graph is K4, whereas the list can be complete only if the intended entry is K1∨\\overline{K3}=K1,3. Please replace this paragraph with an explicit derivation of the list from Lemma 5, δ(F)=2 and e(F)≤10, and add a table or short case argument that exhibits an independent 2-cut in every configuration, including all possible cross-edge distributions.","section":"Section 2, Proof of Theorem 3, basis step (n=7)"},{"comment":"The δ(M)=3 part of the basis step is also asserted: after reducing to M[S]=C3, the paper states that for M[R]∈{K2+K1, P3, C3} 'it is easy to verify that the statement holds by applying Lemma 8 in the latter two cases.' No foresty 3-cut is exhibited, and the application of Lemma 8 is not spelled out for any of the three cases. Since this is the complete n=7 base of the second induction, the proof is not self-contained at the induction base. Please provide the omitted verification, ideally as three explicit constructions of the promised foresty minimum vertex cut or as a short uniform argument using Lemma 8.","section":"Section 2, Proof of Theorem 4, basis step (n=7)"}],"minor_comments":[{"comment":"The claims that Gn has a unique minimum vertex cut in the odd case and exactly two minimum vertex cuts in the even case are stated without proof; the edge count is also stated rather than computed. These assertions are needed for the 'best possible' part of Theorem 3, so a short verification would make the lower-bound construction self-contained.","section":"Section 2, Proof of Theorem 3, sharpness construction"},{"comment":"The statement that κ(Fn)=3 and {v1,v2,v3} is the unique minimum vertex cut is asserted as 'easy to see.' Since this supports the sharpness of the bound in Theorem 4, adding a brief justification would improve the exposition.","section":"Section 2, Proof of Theorem 4, sharpness construction"},{"comment":"The text contains several formatting artifacts, including the missing overline in the expression K1∨K3 and apparent spacing errors such as 'indepen dent' and 'g raph.' If these are LaTeX or OCR artifacts, please ensure the final version uses unambiguous notation, especially for complements.","section":"Throughout"},{"comment":"References [5] and [8] are arXiv preprints; if journal versions have appeared, it would be helpful to update the citations.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"I am sympathetic to acceptance once the two finite basis-step verifications are written out, because the structural arguments in the induction steps and the regular-graph lemmas read as sound. The missing overline in K1∨K3 appears to be a typesetting issue rather than a mathematical error, but it must be fixed because it affects the completeness of the enumeration in the n=7 base of Theorem 3. No concerns about novelty or attribution; the paper is within the scope of math.CO."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this paper deserves a real referee. Theorems 3 and 4 are genuine new results — exact edge thresholds for forcing a minimum vertex cut that is independent (floor(3n/2)) or foresty (2n). The line from Chen-Yu and Le-Pfender to minimum cuts is natural, and Theorem 4 is the first foresty-minimum-cut extremal result I know. The regular-graph lemmas (Lemma 6 for cubic, Lemma 9 for 4-regular) are nontrivial and the induction steps read as sound. The sharpness constructions for odd/even n in Theorem 3 and the F_n family in Theorem 4 look right, based on the stated edge counts and unique minimum cuts.\n\nThe weak spot is exactly where the stress test says: the n=7 bases are asserted, not demonstrated. In Theorem 3, the nine possible shapes for F−N[v] are listed with no derivation, and the notation 'K1∨K3' is ambiguous — if it means the join with a triangle it is K4, which would duplicate K−4; likely an overline was lost and they mean the star K1,3. The list may be correct, but the reader cannot tell from the paper. In Theorem 4, the three cases M[R] ∈ {K2+K1, P3, C3} are disposed of with 'easy to verify' plus an unstated appeal to Lemma 8. That is load-bearing. If any listed case is missing or the claimed cut is not actually a cut, both inductions collapse. A referee should ask for a table or appendix that explicitly exhibits the promised independent/foresty 2-cut or 3-cut for each case.\n\nI should stress that I found no independent error in Lemma 6, Lemma 9, or in the induction arguments themselves. So this is a rigor gap, not a fundamental flaw. The paper is far from a worthless submission; it is a good result with a presentation that is too terse in the one place that matters.\n\nWho is it for: extremal graph theorists and people working on vertex cuts, fragility, and forest cuts. It will be useful as a citation for the new thresholds and the regular-graph lemmas. I would not desk-reject it. Send it to a competent referee with a request to check the base cases carefully. If those checks hold, accept; the author can reasonably be asked to expand the proof.\n\nRecommendation: peer review, yes.","headline":"Solid extremal graph theory paper with two new sharp thresholds; main gap is the hand-waved n=7 base cases, which are load-bearing but likely repairable.","tokens_in":6998,"tokens_out":2529,"would_cite":true,"duration_ms":25353,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C35","05C40","05C69"],"pacs":[],"model":"deepseek-v4-flash","headline":"Every connected graph on $n\\ge 7$ vertices with at most $\\lfloor 3n/2\\rfloor$ edges has an independent minimum vertex cut, and at most $2n$ edges forces a foresty minimum cut; both bounds are best possible.","keywords":["fragile graph","minimum vertex cut","independent vertex cut","foresty vertex cut","vertex connectivity","extremal graph theory","sparse graphs"],"falsifier":"Decisive check: exhaustively enumerate all connected graphs of order 7 with at most 10 edges (and separately at most 14) and verify that every residual graph in the listed base-case classes (the nine possibilities for $F-N[v]$ in Theorem 3 and the three for $M[R]$ in Theorem 4) actually contains the claimed cut; a single miss would disprove the theorem, while a pass would confirm the load-bearing base of the induction.","tokens_in":6036,"feed_emoji":"🔗","tokens_out":16938,"duration_ms":151714,"temperature":0.7,"pith_summary":"The paper proves two exact edge thresholds for structured minimum vertex cuts in connected graphs. A minimum vertex cut is a smallest set of vertices whose removal disconnects the graph; it is independent if the cut vertices share no edges, and foresty if the graph induced on them is a forest. The main theorems say that for every $n\\ge 7$, a connected graph with at most $\\lfloor 3n/2\\rfloor$ edges must have an independent minimum vertex cut, and a connected graph with at most $2n$ edges must have a foresty minimum vertex cut. Both thresholds are best possible: for every $n\\ge 7$ the authors build $n$-vertex graphs with one more edge whose minimum cuts are all an edge, respectively all a triangle. The result matters because it shows that in sparse graphs the connectivity bottleneck cannot be tangled; it must be realized by a structurally simple separator.","feed_headline":"At ⌊3n/2⌋ edges, an independent minimum cut is forced","feed_subtitle":"At 2n edges, a foresty minimum cut is forced. Both bounds are sharp for n≥7.","key_machinery":"The proofs are built on induction over the order $n$, with the induction step splitting according to minimum degree. Lemma 5 is the workhorse: if $S$ is a minimum vertex cut of a connected graph, every vertex of $S$ has a neighbor in every component of $G-S$. Lemma 6 handles cubic graphs, showing every connected cubic graph of order at least 8 has an independent minimum vertex cut, and Lemma 9 handles 4-regular graphs, showing every connected 4-regular graph of order at least 7 has a foresty minimum vertex cut. Lemma 8, derived from Menger's theorem, provides that a component of size at least $k$ must receive a $k$-matching from any vertex cut in a $k$-connected graph. The base case $n=7$ is settled by a finite enumeration of the possible structures left after deleting a degree-2 or degree-3 vertex and its closed neighborhood, with the sharpness examples built as $n$-vertex graphs whose unique minimum cuts are an edge or a triangle.","core_discovery":"The central claim is that sparsity alone forces minimum vertex cuts to be simple. Theorem 3 states that every connected graph of order $n\\ge 7$ with at most $\\lfloor 3n/2\\rfloor$ edges has a minimum vertex cut $S$ whose induced subgraph $G[S]$ is edgeless, and Theorem 4 states that every connected graph of order $n\\ge 7$ with at most $2n$ edges has a minimum vertex cut $S$ with $G[S]$ a forest. The word 'minimum' is essential: earlier results only guaranteed an independent vertex cut of some size under different edge counts, not one of cardinality $\\kappa(G)$. The proofs proceed by induction on $n$, with dedicated lemmas for cubic and 4-regular graphs, and the sharpness is demonstrated by explicit constructions in which every minimum cut induces exactly one edge, respectively exactly one triangle.","pith_inferences":["By the same induction pattern, one can ask for the largest constant $c$ such that every connected $n$-vertex graph with at most $cn$ edges has a minimum vertex cut whose induced subgraph belongs to a fixed hereditary class; the two theorems provide the values $c=3/2$ for edgeless separators and $c=2$ for forest separators.","The finite base case $n=7$ is small enough for a certified computer check; automating that verification would make the proof fully checkable and reusable for other separator classes.","The extremal graphs $G_n$ and $F_n$ have unique or near-unique minimum cuts, which suggests that near the threshold all bad graphs are rigid; classifying the cut structure of all graphs at the critical edge counts would test that rigidity.","A natural next rung is to replace 'forest' with 'graph of bounded treewidth' or 'graph of bounded arboricity' and look for the corresponding edge threshold; the present results are the first two rungs of such a ladder."],"forward_implications":["For every connected $n$-vertex graph with $n\\ge 7$ and at most $\\lfloor 3n/2\\rfloor$ edges, there is a set of exactly $\\kappa(G)$ vertices whose removal disconnects the graph and which induces no edge.","For every connected $n$-vertex graph with $n\\ge 7$ and at most $2n$ edges, there is a minimum vertex cut whose induced subgraph is a forest.","The bounds are best possible: the constructions $G_n$ and $F_n$ show that with one extra edge, a graph can have no independent, respectively foresty, minimum vertex cut.","In the induction, the only graphs requiring separate arguments are cubic graphs and 4-regular graphs; all other cases are reduced to a smaller graph by deleting a degree-2 or degree-3 vertex.","Read together with the earlier fragility results cited in the introduction, the theorems show that the stronger minimum-cut versions still hold with linear edge budgets, namely $\\lfloor 3n/2\\rfloor$ for independent cuts and $2n$ for foresty cuts."],"supporting_citations":[{"why":"Supplies standard terminology and the Menger's theorem statement used in Lemma 7.","marker":"[1]"},{"why":"The earlier fragility theorem that motivates the search for independent cuts and is strengthened by adding minimality.","marker":"[4]"},{"why":"Introduces foresty vertex cuts, the notion whose minimum-cut version Theorem 4 establishes.","marker":"[5]"},{"why":"Characterizes extremal non-fragile graphs at size 2n−3, giving the sharpness context for independent cuts.","marker":"[7]"},{"why":"Provides the (S,T)-path lemma via Menger's theorem used in Lemma 7 and Lemma 8.","marker":"[9]"}],"fun_headline_variants":["For n≥7, 3n/2 edges force an independent minimum cut","For n≥7, 2n edges force a foresty minimum cut","Sparse graphs force simple minimum cuts: independent at 3n/2, foresty at 2n","Minimum vertex cuts become edgeless or foresty in sparse graphs","Sharp bounds: independent min-cut at 3n/2, foresty at 2n for n≥7"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing assumption is the unexpanded finite check at $n=7$: after deleting a degree-2 (or degree-3) vertex and its two (or three) neighbors, the proof lists the possible leftover graphs and asserts that each one contains the required cut; if that list is incomplete or any listed case is wrong, the induction base fails.","fun_headline_variants_meta":{"raw":{"variants":["For n≥7, 3n/2 edges force an independent minimum cut","For n≥7, 2n edges force a foresty minimum cut","Sparse graphs force simple minimum cuts: independent at 3n/2, foresty at 2n","Minimum vertex cuts become edgeless or foresty in sparse graphs","Sharp bounds: independent min-cut at 3n/2, foresty at 2n for n≥7"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001422,"raw_usage":{"total_tokens":5706,"prompt_tokens":875,"completion_tokens":4831,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":491,"completion_tokens_details":{"reasoning_tokens":4716}},"tokens_in":491,"tokens_out":4831,"duration_ms":34236,"temperature":1.0,"reasoning_tokens":4716,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T21:59:27.184778+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Decisive check: exhaustively enumerate all connected graphs of order 7 with at most 10 edges (and separately at most 14) and verify that every residual graph in the listed base-case classes (the nine possibilities for $F-N[v]$ in Theorem 3 and the three for $M[R]$ in Theorem 4) actually contains the claimed cut; a single miss would disprove the theorem, while a pass would confirm the load-bearing base of the induction.","supporting_citations":[{"cited_title":"Bondy and U.S.R","cited_arxiv_id":null,"evidence_quote":"Supplies standard terminology and the Menger's theorem statement used in Lemma 7."},{"cited_title":"Chen and X","cited_arxiv_id":null,"evidence_quote":"The earlier fragility theorem that motivates the search for independent cuts and is strengthened by adding minimality."},{"cited_title":"Le and F","cited_arxiv_id":null,"evidence_quote":"Characterizes extremal non-fragile graphs at size 2n−3, giving the sharpness context for independent cuts."},{"cited_title":"West, Introduction to Graph Theory, Prentice Hall, Inc., 19 96","cited_arxiv_id":null,"evidence_quote":"Provides the (S,T)-path lemma via Menger's theorem used in Lemma 7 and Lemma 8."}],"review_version":1}