{"id":"c8647536-ea13-4e95-90eb-5f737127c747","arxiv_id":"1908.04340","paper_version":8,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Any finite connected graph with edges labeled 0 or 1 and obeying simple balance rules is the Reeb graph of a smooth function on an orientable surface or higher-dimensional manifold, with level sets prescribed as circles or lines, spheres or Euclidean spaces.","lead":"A topologist proves that many finite graphs can be realized as the Reeb graph of a smooth function on a surface or a higher-dimensional manifold, with the type of every level set chosen in advance. The construction works on open manifolds as well as closed ones, and it allows non-compact level sets such as lines and Euclidean spaces.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The unbalanced-case theorems rest on asserted but unconstructed fold maps F0, F, F′, F1, and Theorems 3 and 5 explicitly defer proofs, so the full realizability claim is not yet established.","rationale":"The reader’s weakest assumption identifies the same point: the proof of Theorem 4 assumes the existence of smooth fold maps F0, F, F′, F1 with specific singular value sets and fiber structures, without constructing them. This is the most load-bearing concern because Theorem 4 is the mechanism that extends realizability from the balanced cases to all cases where the balance conditions fail, and Theorem 5 plus parts of Theorem 6 are built on it. The manuscript itself flags that Theorems 3 and 5 leave rigorous proofs to the reader, which further supports treating the full claim as conditional. I do not see a demonstrated error in Theorem 2, whose handle-attachment argument is plausible, and the asserted fold maps are likely standard objects; but until their existence and the deferred higher-dimensional proofs are independently checked, a conditional verdict is appropriate. The concrete test of constructing F0 and verifying its properties would settle whether the concern lands. If the construction goes through as stated, the unbalanced theorems can be accepted modulo the deferred higher-dimensional details; if it fails, the theorem needs revision. Therefore I recommend keeping the reader’s CONDITIONAL verdict unchanged.","tokens_in":14026,"tokens_out":46979,"duration_ms":505788,"concrete_test":"Write out an explicit construction of F0 in CASE 1-A, for instance by taking a suitable double cover of the region D0 branched over the prescribed singular-value arcs, and verify all listed properties: that the map is a smooth fold map, the singular set maps by an embedding to the stated curves, the image and singular-value sets agree with the description, and the composition with p has singular value set {0} with regular fibers of the claimed topology. If this succeeds, repeat the same verification for F, F′, and F1. If any one of these maps cannot be constructed with the stated properties, identify which balance-defect class fails; this would directly falsify or restrict Theorem 4.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing gap is in Section 3, Theorem 4, STEP 1. For each unbalanced local configuration (CASE 1-A through 1-D) the proof begins “We can construct a smooth map F0/F/F′/F1 from a surface into the plane satisfying...” and then lists detailed properties about the singular value set, the image, and the preimage structure. No construction or verification is supplied. These maps are not auxiliary: the desired function in Theorem 4 is obtained by composing them with the projection p(x,y):=y and scaling, and Theorem 5 is the same construction in higher dimensions. If any of these fold maps cannot be realized with all stated properties, then the unbalanced-case statements (and the local replacements in Theorem 6) do not follow. Additionally, Theorem 3 and Theorem 5 explicitly say “Rigorous proofs are left to readers,” so the higher-dimensional balanced and unbalanced claims are only sketched. The balanced surface case (Theorem 2) is more explicit, but even its final gluing/orientability step is asserted rather than fully demonstrated. The concern is about completeness, not a demonstrated contradiction; nevertheless it is load-bearing because the paper’s advertised answer to the realizability problem includes these cases.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the realizability problem for Reeb graphs: given a finite connected graph G, a good function h, and edge labels in {0,1} (or nonnegative integers in Theorem 6), the author seeks a connected orientable surface or manifold M and a smooth function f whose Reeb graph is isomorphic to G, with regular preimages prescribed by the labels. Theorem 2 gives a positive result for surfaces under balance conditions: at non-extremal vertices the numbers of incoming and outgoing label-1 edges agree, with an extra condition when that number is 1, and at extremal vertices the number of incident label-1 edges is even; the fibers are circles for label 0 and lines for label 1. Theorem 3 states the analogous higher-dimensional statement with S^n and R^n fibers. Theorems 4 and 5 treat the unbalanced cases, where the balance conditions fail, using local fold-map models; Theorem 6 gives a further higher-dimensional construction with nonnegative edge labels and fibers obtained from S^n by deleting discs. The proofs are constructive in outline: local functions around vertices are built by handle attachments or by composing fold maps with the projection, then glued along boundary components.","tokens_in":14351,"tokens_out":4344,"duration_ms":45999,"significance":"If the missing details are supplied, the paper would make a substantial contribution to the Reeb-graph realization problem. It extends prior work by Sharko, Masumoto-Saeki, Michalak, and the author's own 3-dimensional construction to open manifolds and to functions with more general singularities, and it states precise combinatorial conditions that are plausibly near-optimal for the prescribed-fiber version of the problem. The approach is direct and self-contained, with no circularity or fitted parameters; the balanced surface case is credible because the handle-attachment construction is described in explicit local models. The main weakness is completeness: Theorem 3 and Theorem 5 are explicitly left to the reader, and Theorem 4 relies on asserted local fold maps without construction or verification. The significance of the results is therefore conditional on completing those proofs.","major_comments":[{"comment":"The proof of Theorem 4 asserts the existence of smooth fold maps F0, F, F', and F1 with specified singular value sets, images, collar behaviors, and preimage structures, but no construction or verification is supplied. These maps are load-bearing: the desired local function is obtained by composing them with the projection p(x,y):=y and scaling, and the same models are invoked in the higher-dimensional Theorem 5 and in parts of Theorem 6. Without an explicit construction or a reference proving that such fold maps exist, the unbalanced-case theorems are not established. This is not a cosmetic gap but the central mechanism of the proof.","section":"Section 3, STEP 1 (CASE 1-A through 1-D)"},{"comment":"Theorem 3, which is stated as a higher-dimensional version of Theorem 2, is not proved. The text says 'Rigorous proofs are left to readers.' Remark 1 sketches handle attachments for the non-extremal vertex case, but it does not treat the extremal-vertex cases (Case 1 and Case 2) or the gluing and orientability step in dimension n+1. Since Theorem 3 is one of the paper's main advertised results, a complete proof or at least a fully detailed proof outline covering all cases must be included.","section":"Section 2, after Remark 1"},{"comment":"Theorem 5, the (n+1)-dimensional unbalanced analogue of Theorem 4, is stated with only a remark and the sentence 'rigorous proofs are left to readers.' The single remark that 'we take a standard (n-1)-dimensional sphere instead of the two point set' is not a proof. Because Theorem 5 is a central result of the paper and depends on the unproved fold-map constructions of Theorem 4, its omission is a load-bearing gap.","section":"Section 3, Theorem 5"},{"comment":"The final gluing step states that gluing the local functions together on the boundary 1-manifolds gives a desired function on a surface, and that 'to make the resulting surface orientable, we must use the diffeomorphisms for the gluing carefully one after another.' Since orientability is part of the theorem's conclusion, the proof should specify the gluing diffeomorphisms and explain why they can be chosen consistently at all boundary components, including at vertices of degree greater than 2 and across edges with both label 0 and label 1. This gap is likely fixable, but as written it is a missing argument in an otherwise explicit construction.","section":"Section 2, STEP 3 of the proof of Theorem 2"}],"minor_comments":[{"comment":"There are several typographical errors: 'gra ph' in the abstract, 'uniquey' in Definition 1, and inconsistent spacing in the author name in the header. These should be corrected.","section":"Abstract and throughout"},{"comment":"The phrasing 'the number of edges 1's are assigned to and containing the vertex as the starting points' is grammatically confusing. It would be clearer to define, for each vertex, the number of incident label-1 edges oriented out of the vertex and the number oriented into the vertex, and then state the balance condition in terms of these numbers.","section":"Section 1, Definition and Theorem 2"},{"comment":"The caption of Figure 3 contains unclear wording: 'for a n-handles' and 'to which the n-handles are attached' are awkward, and the figure omits many handles described in Remark 1. A more detailed caption or a more complete figure would help the reader follow the higher-dimensional handle attachments.","section":"Figure 3 caption"},{"comment":"The condition 'if an edge contains a vertex of degree 1, then 0, 1 or 2 is assigned to the edge' is stated without explaining whether an edge with two degree-1 vertices (the single-edge graph) is allowed and what the fiber condition means in that case. This should be clarified.","section":"Section 4, Theorem 6"},{"comment":"Reference [4] is cited as 'arxiv:1901.04994v1' without indication of whether it has been published or revised; if a journal version exists, it should be cited. Also, the paper would benefit from more complete references to the recent realizability results mentioned in the introduction.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of the journal and addresses a natural problem in differential topology. The author's approach is promising and the balanced surface case is nearly complete, but two main theorems (Theorems 3 and 5) are explicitly left without proofs, and Theorem 4 relies on asserted local fold maps. In my view these are fixable within the manuscript's scope, but the current version cannot be accepted as is. I recommend major revision, with the expectation that the author supply rigorous constructions or precise references for the missing local models and complete the omitted proofs."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the paper is worth reading for Theorem 2, the balanced surface realizability with circle and line fibers. The rest of the advertised program is not yet proven.\n\nWhat is new: the surface case allowing non-compact fibers (line components), the higher-dimensional version with S^n/R^n fibers, the treatment of graphs that violate the balance conditions, and the punctured-sphere fiber theorem. These are real extensions of the earlier work by Masumoto–Saeki, Michalak, and the author's own 3-dimensional result. The proof of Theorem 2 is given in reasonable detail via handle attachments, and the local picture around vertices is convincing. The author also cites prior work fairly; self-citations are for method, not as unproved inputs.\n\nThe soft spots are load-bearing. Theorem 3 and Theorem 5 explicitly say rigorous proofs are left to readers. That alone would be fine for a remark or conjecture, but here these are central advertised theorems. More concerning is Theorem 4, STEP 1: in each case the proof asserts \"We can construct a smooth map F0/F/F'/F1\" with a long list of precise properties about the singular value set, the image, and the preimages, and then gives no construction or verification. These fold maps are the mechanism that makes the unbalanced cases work; if they cannot be realized exactly as stated, Theorems 4 and 5 do not follow. The same local models are used in Theorem 6. I also agree with the stress-test note that the orientability step in Theorem 2's gluing is only described in words, though that is a minor gap compared with Section 3.\n\nI want to be clear: I do not see a demonstrated error. The constructions are plausible and are in the spirit of the author's earlier work. But the paper's own text admits the higher-dimensional proofs are sketches, and the unbalanced case hangs on unproved existence assertions. That is not a complete answer to the realizability problem as stated.\n\nWho is this for? Researchers working on Reeb graphs and singularity theory who want the balanced surface result and a clear map of what remains open. The paper deserves a serious referee: the question is natural, the balanced case looks close to correct, and the gaps are addressable. If it lands on my desk I would send it to review with the expectation of major revision, pressing the author to either prove the asserted fold maps or restate the theorems to match what is actually shown.","headline":"The balanced surface case is a genuine, plausible result; the higher-dimensional and unbalanced theorems are not yet proven and rely on asserted local fold maps.","tokens_in":14766,"tokens_out":1943,"would_cite":false,"duration_ms":24201,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["57R45","57R19"],"pacs":[],"model":"deepseek-v4-flash","headline":"Every balanced 0-1 labeled graph arises as a Reeb graph","keywords":["Reeb graph","smooth function","Morse function","Morse-Bott function","fold map","realization problem","orientable surface","singularities of differentiable maps"],"falsifier":"Check the asserted fold maps in CASE 1-A through 1-D of Section 3: attempt to write down or cite a standard construction of a smooth fold map from an open surface into the plane whose singular value set is, for example, the union of a half-line, a point, a decreasing graph, and a parabolic arc (CASE 1-B) and whose regular fibers over the complement are two points, with the specified product structure near the singular arcs. If any of these local pictures is not realizable, Theorem 4's STEP 1 fails; conversely, an explicit coordinate model would confirm them and make the unbalanced construction fully constructive.","tokens_in":13854,"feed_emoji":"🔗","tokens_out":12248,"duration_ms":99506,"temperature":0.7,"pith_summary":"This paper proves a constructive realization statement for Reeb graphs: any finite connected graph, with each edge labeled 0 or 1 and satisfying two local balance conditions, appears as the Reeb graph of a smooth function on a connected orientable surface (Theorem 2), and in all higher dimensions on an orientable $(n+1)$-manifold (Theorem 3). Along a 0-edge the preimage of a regular value is a circle (or $S^n$); along a 1-edge it is a line (or $\\mathbb{R}^n$), and the function takes the prescribed values at vertices. The paper then shows the balance conditions are not essential: if they fail, the same conclusion holds with finitely many exceptional singular points (Theorems 4 and 5), and for $n>1$ edges may carry arbitrary non-negative labels, with regular preimages diffeomorphic to $S^n$ with that many disjoint open discs removed, as long as degree-one edges carry 0, 1, or 2 (Theorem 6).","feed_headline":"Every balanced 0-1 labeled graph arises as a Reeb graph","feed_subtitle":"The construction works for open surfaces and all higher dimensions, allowing line fibers instead of only circle fibers.","key_machinery":"The engine is a local-to-global handle construction. Around each vertex of $G$, the paper builds a local Morse (or Morse-Bott/fold) function on a model surface whose Reeb graph is a small neighborhood of that vertex: one starts with the disjoint union of $a$ lines and $b$ circles (for the relevant triples $(a,b,c)$) crossed with an interval, attaches 1-handles (and, in dimension $n+1$, $n$-handles) to connect the components, and obtains a surface whose boundary has the required numbers of line and circle components. These local models are then glued along trivial bundles over the edges. For the cases where the balance conditions fail, the paper introduces explicit fold maps $F_0, F, F', F_1$ from open surfaces into the plane whose image is bounded by graphs of smooth functions; composing with the projection to the $y$-axis produces a function with a single singular value that changes the number of line components in the regular fibers, repairing the imbalance.","core_discovery":"The central claim is that the realizability of a graph as a Reeb graph with prescribed regular fibers is governed by simple integer conditions at vertices. Given a good function $h$ on a finite connected graph $G$ and labels $0$ or $1$ on the edges, the conditions are: at a vertex where $h$ has no local extremum, the number of 1-labeled edges starting at the vertex equals the number ending there, with the extra proviso that if this number is 1 then some 0-labeled edge also meets the vertex; at a local extremum, the number of 1-labeled edges is even. Under these conditions, Theorem 2 produces a connected orientable surface $M$ and a smooth function $f : M \\to \\mathbb{R}$ whose Reeb graph is isomorphic to $G$, with regular preimages circles on 0-edges and lines on 1-edges, and with $f$ matching $h$ at the vertices; Theorem 3 gives the same statement with $S^n$ and $\\mathbb{R}^n$ fibers on an orientable $(n+1)$-manifold. The singularities are controlled to be Morse at non-extremal vertices and Morse, Morse-Bott, or compositions of two Morse functions at extremal vertices.","pith_inferences":["Beyond the paper's claims, the balance conditions read as a conservation law for 1-labeled (non-compact) edges through each vertex; testing whether they are also necessary for realization with exactly these fibers and singularity classes would settle the sharpness of the theorems.","The fold maps $F_0, F, F', F_1$ in Section 3 are specified by their images and fiber structures; giving explicit coordinate formulas for them would make the unbalanced construction fully concrete and would allow computation of the resulting surface's genus.","Theorem 6 restricts degree-one edge labels to 0, 1, or 2; an extension not attempted here would be to decide whether label 3 can be realized if more general fold singularities are permitted.","Because 1-edges correspond to non-compact fibers, the surfaces produced when any 1-edge is present are necessarily open; spelling out how the number of 1-edges relates to the surface's ends would connect this work to the topology of non-compact surfaces."],"forward_implications":["Every finite connected graph with 0/1 labels satisfying the two balance conditions is realizable on some orientable surface, so the only local obstructions to Reeb-graph realization with circle/line fibers are these numerical conditions.","The same conditions work in every dimension: for each $n \\ge 1$, an orientable $(n+1)$-manifold admits a function whose Reeb graph is the given graph with $S^n$ and $\\mathbb{R}^n$ regular fibers.","Failure of the balance conditions does not prevent realization: only finitely many exceptional singular points of fold type are needed, and the regular fiber types are unchanged.","For $n>1$, edge labels can be arbitrary non-negative integers (with degree-one edges restricted to 0, 1, or 2), yielding regular preimages diffeomorphic to $S^n$ with that many disjoint open discs removed.","The construction respects the prescribed ordering of critical values, since the function takes the given good function's values at vertices."],"supporting_citations":[{"why":"Launches the problem of realizing a given graph as the Reeb graph of a smooth function on a manifold, which this paper's theorems answer in a generalized setting.","marker":"[14]"},{"why":"Supplies the author's earlier construction for 3-dimensional closed orientable manifolds with arbitrary surface fibers, whose handle-attachment method is adapted and extended here to all dimensions and open manifolds.","marker":"[5]"},{"why":"Gives prior realizability results for smooth functions on manifolds with given Reeb graph, restricted to closed surfaces or sphere-fiber settings that this paper goes beyond.","marker":"[7]"},{"why":"Proves realization of a graph as the Reeb graph of a Morse function on a manifold, providing the baseline Morse-theoretic handles used in the local constructions.","marker":"[8]"},{"why":"Studies combinatorial modifications of Reeb graphs and the realization problem, giving context for which graphs are known to be realizable and how local changes can be performed.","marker":"[9]"}],"fun_headline_variants":["Simple vertex rules decide if a graph is a Reeb graph","Reeb graphs with line fibers: new realizability theorem","Open manifolds included: Reeb graphs from simple conditions","When are graphs Reeb graphs? Now answered for open manifolds","Integer vertex conditions fully characterize Reeb graph realizability"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof of the unbalanced cases (Theorems 4 and 5, and the local fix in Theorem 6) assumes that smooth fold maps $F_0, F, F', F_1$ exist with the specific singular value sets, images, and preimage structures described in Section 3, but these maps are asserted rather than explicitly constructed; if they cannot be built as stated, those theorems do not follow.","fun_headline_variants_meta":{"raw":{"variants":["Simple vertex rules decide if a graph is a Reeb graph","Reeb graphs with line fibers: new realizability theorem","Open manifolds included: Reeb graphs from simple conditions","When are graphs Reeb graphs? Now answered for open manifolds","Integer vertex conditions fully characterize Reeb graph realizability"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000816,"raw_usage":{"total_tokens":3618,"prompt_tokens":1034,"completion_tokens":2584,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":650,"completion_tokens_details":{"reasoning_tokens":2501}},"tokens_in":650,"tokens_out":2584,"duration_ms":17165,"temperature":1.0,"reasoning_tokens":2501,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:45:22.136600+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check the asserted fold maps in CASE 1-A through 1-D of Section 3: attempt to write down or cite a standard construction of a smooth fold map from an open surface into the plane whose singular value set is, for example, the union of a half-line, a point, a decreasing graph, and a parabolic arc (CASE 1-B) and whose regular fibers over the complement are two points, with the specified product structure near the singular arcs. If any of these local pictures is not realizable, Theorem 4's STEP 1 fails; conversely, an explicit coordinate model would confirm them and make the unbalanced construction fully constructive.","supporting_citations":[{"cited_title":"Sharko, About Kronrod-Reeb graph of a function on a manifold , Methods of Functional Analysis and Topology 12 (2006), 389–396","cited_arxiv_id":null,"evidence_quote":"Launches the problem of realizing a given graph as the Reeb graph of a smooth function on a manifold, which this paper's theorems answer in a generalized setting."},{"cited_title":"Combinatorial modifications of Reeb graphs and the realization problem","cited_arxiv_id":"1811.08031","evidence_quote":"Studies combinatorial modifications of Reeb graphs and the realization problem, giving context for which graphs are known to be realizable and how local changes can be performed."}],"review_version":1}