{"id":"30319539-a7b3-4188-8743-4586ed788c13","arxiv_id":"1908.03769","paper_version":3,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For a graph G and any of its splitting graphs G', the paper proves inequalities such as reg(I(G)) <= reg(I(G')) and proj dim(I(G)) <= proj dim(I(G')) for special splittings and many graph classes.","lead":"This paper introduces a new graph operation called splitting, where one graph maps onto another while matching edges one-to-one, and compares algebraic invariants of the associated edge ideals. It proves that for special classes of splittings, projective dimension and regularity of the original graph are bounded by those of the split graph.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; the only soft spot is the terse induction step in Theorem 1.3, which is fillable.","rationale":"I re-read the central theorem and its proof. The most load-bearing step is the unstated preservation of the 'special' property under iterated vertex identifications in Theorem 1.3. The reader flagged this as the weakest assumption, and I agree that it is the place to probe. Working through the one-step contraction, the induced map is indeed a splitting map, and condition (1) is preserved in all cases. The only nontrivial case is when a remaining fiber has one vertex outside the neighborhood of the deleted vertex y and one inside it; the original special condition then forces the edges needed after contraction. So the proof can be completed without changing the theorem. I also checked Lemma 1.4, the short exact sequence, the regularity and projective dimension inequalities, and the component-wise argument for condition (2); the apparent ring abuse in condition (2) is a notational shortcut that can be made rigorous via pd invariance under polynomial extension and the disjoint-variable tensor product. Propositions 1.5 and 1.6 are standard and their proofs are sound once one notes that the ν in Proposition 1.6 is the induced matching number, as the proof and cited results indicate. No fatal gap or false claim emerged, so the reader's ACCEPT verdict should stand unchanged.","tokens_in":8712,"tokens_out":57077,"duration_ms":628936,"concrete_test":"State and prove the missing preservation lemma: for a special splitting map f: H -> G satisfying condition (1) and vertices x,y with f(x)=f(y), the contracted graph H' with the induced vertex map is again a special splitting graph of G. Check the three cases for a remaining same-image pair (u,u'): (i) neither is in N_H(y); (ii) exactly one is in N_H(y); (iii) one is x. In case (ii), if a is a neighbor of the vertex outside N_H(y), specialness of that vertex with the vertex inside N_H(y) forces a to be adjacent to y, hence x-a is an edge of H'. Completing this case analysis settles the induction in Theorem 1.3.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest assumption is exactly the softest point: the proof of Theorem 1.3(i)-(ii) for condition (1) defines a sequence G' = G0, G1, ..., Gt = G of single vertex identifications and asserts that each Gi-1 is a special splitting graph of Gi 'with the same argument.' This preservation lemma is not proved. However, it is true. If H is a special splitting graph of G with splitting map f satisfying condition (1), and x,y satisfy f(x)=f(y), then the graph H' obtained by deleting y and joining x to every old neighbor of y is again a special splitting graph of G with the induced map. The delicate case is a remaining pair u,u' with f(u)=f(u') where exactly one of them, say u', lies in N_H(y). For a neighbor a of u, the special condition applied to u,u' forces a to be adjacent to y (because y is a neighbor of u'); hence a is among the new neighbors of x, so the edge x-a is present in H'. The other cases reduce directly to the original special condition. Thus the omission is an exposition gap, not a mathematical flaw. The ring-theoretic shorthand in the condition (2) proof can also be made rigorous by passing through the component rings and using invariance of projective dimension under polynomial extension. The central claim therefore stands under scrutiny.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces the notion of a splitting graph of a given graph G, defined by a surjective vertex map whose induced edge map is bijective. The central result is Theorem 1.3, which states that if G' is a special splitting graph of G (satisfying one of two explicit conditions on fibers of the splitting map), then proj dim(I(G)) ≤ proj dim(I(G')) and reg(I(G)) ≤ reg(I(G')). The proof uses a colon-ideal lemma (Lemma 1.4) and a short exact sequence associated with identifying two vertices. The authors then give partial results for arbitrary splittings: regularity comparison when G is chordal, weakly chordal, sequentially Cohen-Macaulay bipartite, unmixed bipartite, very well-covered, or C5-free vertex decomposable (Proposition 1.6); projective dimension comparison when pd equals bight (Proposition 1.5) or when the splitting is special of type (2) (Proposition 1.7); dimension comparison for all graphs and depth comparison for paths and even cycles (Proposition 1.8); and closure of the bipartite and forest classes under arbitrary splittings (Propositions 1.10 and 1.11). The final section connects splitting graphs to the stretching operator from algebraic shifting theory.","tokens_in":8988,"tokens_out":30640,"duration_ms":307207,"significance":"If the result holds, the paper provides a new structural tool for comparing homological invariants of edge ideals under vertex identifications, a construction that appears naturally in algebraic shifting. The central theorem is substantive and the supporting lemmas are mostly self-contained; Lemma 1.4 is a clean and useful colon computation. The paper also gives explicit, falsifiable consequences for several well-studied graph classes, and the induced-matching lifting argument in Proposition 1.6 is elegant. The main claims are not obtained by curve fitting or circular reasoning; external citations are used only as genuine lower bounds. The main weaknesses are local exposition gaps rather than mathematical errors.","major_comments":[],"minor_comments":[{"comment":"The assertion that one can iterate the vertex-identification construction to obtain a sequence G' = G0, G1, ..., Gt = G with each Gi-1 a special splitting graph of Gi is not proved. This is a necessary induction step: after identifying x and y, one must verify that the new graph G1 still satisfies condition (1) of Definition 1.2 for all remaining pairs with equal image. In particular, the case where exactly one member of a remaining pair lies in NG0(y) requires a short argument. I believe the claim is true, but the proof should be supplied for completeness.","section":"Theorem 1.3, proof (i), condition (1)"},{"comment":"The paragraph for condition (2) switches between the rings S and S' without making the identifications explicit. The ideal I(G) lives in S, not S', and the statement 'I(G'i) = I(Gi)' is an isomorphism after relabeling variables, not an equality. The projective dimension formulas should be written carefully, passing through the component rings and using the fact that projective dimension is invariant under polynomial extension when the variable sets are disjoint.","section":"Theorem 1.3, proof (i), condition (2)"},{"comment":"The sentence 'The argument for cycles of even length is similar' omits the whole proof for the even-cycle case. Since the depth inequality for even cycles is explicitly claimed, please either provide the argument or state the relevant projective dimension formulas for cycle graphs that allow the path-graph argument to be repeated.","section":"Proposition 1.8"},{"comment":"When applying Lemma 1.4 to conclude (I : x-y) = I, the paper does not state why the hypothesis NG0[x] ∩ NG0[y] = ∅ holds. It follows from condition (1) and the fact that the edge map is bijective, but this should be mentioned for clarity.","section":"Theorem 1.3, proof (i), usage of Lemma 1.4"},{"comment":"The paper alternates between proj dim(I(G)) and proj dim(S/I(G)) without comment. These quantities differ by 1 for nonzero proper ideals. Please make the notation consistent in the statements of Theorem 1.3, Propositions 1.5 and 1.7, and the introduction.","section":"Notation throughout"},{"comment":"There are minor typos: in the abstract, 'edge ideals of graph s' should read 'edge ideals of graphs', and in the discussion of the stretching operator, 'transforms u into a squarefree monomial ideal' should read 'squarefree monomial'.","section":"Abstract and final section"}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThis paper introduces splitting graphs and proves that for special splittings, projective dimension and regularity are monotone: proj dim(I(G)) ≤ proj dim(I(G')) and reg(I(G)) ≤ reg(I(G')). That is a useful comparison tool, and the connection to Kalai's stretching operator gives it a natural home. The concept is new, and the main theorem is not a repackaging of known inequalities. Lemma 1.4 is carefully proved and is the right engine for the argument. Propositions 1.5–1.7 sensibly leverage known characterizations of graphs with extremal regularity or projective dimension, and the vertex-cover argument in 1.8 is clean. The paper is also honest that the general case for arbitrary splittings remains open, and the counterexample to a Betti-number inequality is helpful.\n\nSoft spots are what you might expect from a paper that says “the others can be proved in the same way.” The induction in Theorem 1.3 for condition (1) asserts, rather than proves, that iterated vertex identifications preserve the special splitting property. I checked the one-step preservation; it works, with the delicate case being a shared neighbor of the two identified vertices. So the gap is fillable. The condition (2) argument silently changes polynomial rings; you need a sentence about extending scalars or passing to component rings, but the conclusion is right. Proposition 1.8's even-cycle case is dismissed as “similar” without detail—minor, but sloppy, especially since the path calculation itself has a few loose inequalities. There are small typos in Lemma 1.4's proof, but they do not affect the argument.\n\nNone of this undermines the main results. The paper deserves peer review, and a good referee should ask for the preserved-special lemma to be written out and for the even-cycle argument to be filled in. Both are doable.\n\nBottom line: a solid, citable contribution to edge ideal theory and shifting. I would bring it to a reading group and would cite it in work on graph invariants.\n\nBest,\n[Your name]","headline":"A genuinely new graph construction with a correct central inequality; the proof gaps are fillable exposition, not mathematical flaws.","tokens_in":9461,"tokens_out":2839,"would_cite":true,"duration_ms":30967,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13F20","13H10"],"pacs":[],"model":"deepseek-v4-flash","headline":"A splitting graph is a graph whose edges match a given graph while some vertices are duplicated, and this paper proves that for special splittings the edge ideal of the original graph has projective dimension and regularity no larger than…","keywords":["edge ideals","splitting graphs","projective dimension","Castelnuovo-Mumford regularity","vertex identification","Betti numbers","monomial ideals","graph operations"],"falsifier":"Enumerate all special splitting graphs on at most eight vertices and directly compute $\\operatorname{proj\\,dim}$ and $\\operatorname{reg}$ of the two edge ideals over the rationals; any pair violating $\\operatorname{proj\\,dim}(I(G)) \\le \\operatorname{proj\\,dim}(I(G'))$ or $\\operatorname{reg}(I(G)) \\le \\operatorname{reg}(I(G'))$ would falsify Theorem 1.3. A more surgical check is to perform the paper's iterative identifications and test at each step whether $I:(x-y)=I$; the first failure would pinpoint where the proof breaks.","tokens_in":8561,"feed_emoji":"🧩","tokens_out":13771,"duration_ms":137483,"temperature":0.7,"pith_summary":"Given a graph $G$, a splitting graph $G'$ is a graph obtained by letting the edges of $G$ correspond bijectively to edges of $G'$ while some vertices of $G'$ map to the same vertex of $G$. The paper asks whether the edge ideal of $G$ is homologically simpler than the edge ideal of $G'$: specifically, whether $\\operatorname{proj\\,dim}(I(G)) \\le \\operatorname{proj\\,dim}(I(G'))$ and $\\operatorname{reg}(I(G)) \\le \\operatorname{reg}(I(G'))$. The paper proves these two inequalities for special splitting graphs, where identified vertices either have fully cross-adjacent neighborhoods or lie in different connected components. It also proves the inequalities for broad graph classes such as sequentially Cohen-Macaulay, chordal, weakly chordal, unmixed bipartite, very well-covered, and $C_5$-free vertex decomposable graphs. The upshot is that a purely combinatorial operation on graphs, also produced by the stretching operator from shifting theory, has controlled consequences for the homological invariants of edge ideals.","feed_headline":"Fusing split vertices cannot raise regularity or projective dimension","feed_subtitle":"For special graph splittings, the original graph's edge ideal has no larger regularity or projective dimension than the split one.","key_machinery":"The load-bearing object is the special splitting map, and the load-bearing identity is Lemma 1.4: for vertices $x,y$ with disjoint closed neighborhoods, $I(G):(x-y) = I(G) + (zw : z \\in N_G(x), w \\in N_G(y))$. Under special condition (1), the extra generators disappear because every neighbor of $x$ is adjacent to every neighbor of $y$, so the colon ideal equals $I(G)$. Substituting this equality into the short exact sequence yields both inequalities, since $x-y$ behaves as a nonzero divisor modulo the ideal of the graph obtained by identifying $x$ and $y$. Under special condition (2), the splitting graph splits into components and the same conclusion follows from the behavior of projective dimension and regularity for sums of ideals in disjoint variable sets.","core_discovery":"The paper's central discovery is Theorem 1.3: if $G'$ is a special splitting graph of $G$, then $\\operatorname{proj\\,dim}(I(G)) \\le \\operatorname{proj\\,dim}(I(G'))$ and $\\operatorname{reg}(I(G)) \\le \\operatorname{reg}(I(G'))$. A special splitting map is one in which any two vertices of $G'$ with the same image either have every neighbor of one adjacent to every neighbor of the other, or belong to different connected components of $G'$. The proof repeatedly identifies two such vertices and uses the short exact sequence $0 \\to (S'/(I:x-y))(-1) \\to S'/I \\to S'/(I,x-y) \\to 0$ to transfer the two inequalities from each identification step to the next. The paper further obtains the projective dimension inequality for sequentially Cohen-Macaulay graphs and the regularity inequality for several families in which regularity is governed by the matching number.","pith_inferences":["If the expected inequalities hold for all splittings rather than only special ones, then edge-preserving vertex identification of this kind would be a homological degeneration: projective dimension and regularity would be upper semicontinuous under contraction. A systematic computer search over all graphs with at most seven vertices could test this before any proof is attempted.","The paper's example showing that individual Betti numbers can decrease under splitting suggests that the true general statement, if one exists, concerns total Betti numbers or extremal invariants rather than componentwise comparison; comparing total Betti numbers across all small splittings is a concrete next experiment.","Because $\\sigma$-stable graphs are splitting graphs produced by the stretching operator, determining whether those particular splittings are special, or proving the inequalities directly for them, would connect Theorem 1.3 to algebraic shifting and sharpen the picture of how stretching affects resolutions of edge ideals."],"forward_implications":["For every special splitting graph, the original graph's edge ideal has projective dimension and regularity no larger than the splitting graph's, so fusing such duplicated vertices cannot create homological complexity.","If $G$ is sequentially Cohen-Macaulay, the projective dimension inequality holds for every splitting graph, because projective dimension equals the largest minimal vertex cover size and vertex covers lift to the splitting graph.","If $G$ is chordal, weakly chordal, sequentially Cohen-Macaulay bipartite, unmixed bipartite, very well-covered, or $C_5$-free vertex decomposable, the regularity inequality holds for every splitting graph, since in these families the regularity is $\\nu(G)+1$ and induced matchings lift to the splitting graph.","For special splittings satisfying condition (2), the graded Betti number inequality $\\beta_i(I(G)) \\le \\beta_i(I(G'))$ holds for all $i$.","For every graph and every splitting graph, $\\dim(S'/I(G')) \\ge \\dim(S/I(G))$, and for paths and even cycles the same monotonicity holds for depth."],"supporting_citations":[{"why":"Supplies the sum-of-ideals bound on projective dimension used in the connected-components case of special splittings.","marker":"[3]"},{"why":"Gives the regularity bound for sums of ideals that motivates the expected inequalities for splitting graphs.","marker":"[7]"},{"why":"Provides the inequality $\\operatorname{bight}(I(G'))-1 \\le \\operatorname{proj\\,dim}(I(G'))$ used to prove projective dimension monotonicity for sequentially Cohen-Macaulay graphs.","marker":"[12]"},{"why":"Provides $\\nu(G')+1 \\le \\operatorname{reg}(I(G'))$, the matching-number lower bound that transfers the regularity inequality to the listed graph families.","marker":"[8]"}],"fun_headline_variants":["Splitting graphs can't drop reg or proj dim","Fusing split vertices: no reg or pd increase","Special splits: original's reg and pd ≤ split's","Original invariants bounded above by special split's","Original edge ideal's reg and pd never exceed split's"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes that when two vertices lying over the same vertex are fused, and then the next such pair is fused, and so on, the two vertices being fused never acquire a common neighbor, so that the equality $I:(x-y)=I$ continues to hold at every step; this preservation is asserted rather than proved.","fun_headline_variants_meta":{"raw":{"variants":["Splitting graphs can't drop reg or proj dim","Fusing split vertices: no reg or pd increase","Special splits: original's reg and pd ≤ split's","Original invariants bounded above by special split's","Original edge ideal's reg and pd never exceed split's"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001358,"raw_usage":{"total_tokens":5411,"prompt_tokens":749,"completion_tokens":4662,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":365,"completion_tokens_details":{"reasoning_tokens":4583}},"tokens_in":365,"tokens_out":4662,"duration_ms":42429,"temperature":1.0,"reasoning_tokens":4583,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:07:31.472028+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Enumerate all special splitting graphs on at most eight vertices and directly compute $\\operatorname{proj\\,dim}$ and $\\operatorname{reg}$ of the two edge ideals over the rationals; any pair violating $\\operatorname{proj\\,dim}(I(G)) \\le \\operatorname{proj\\,dim}(I(G'))$ or $\\operatorname{reg}(I(G)) \\le \\operatorname{reg}(I(G'))$ would falsify Theorem 1.3. A more surgical check is to perform the paper's iterative identifications and test at each step whether $I:(x-y)=I$; the first failure would pinpoint where the proof breaks.","supporting_citations":[{"cited_title":"Herzog, A generalization of the Taylor complex construction, Communications in Algebra 35.5 (2007), 1747–1756","cited_arxiv_id":null,"evidence_quote":"Supplies the sum-of-ideals bound on projective dimension used in the connected-components case of special splittings."},{"cited_title":"Intersections of Leray complexes and regularity of monomial ideals","cited_arxiv_id":"math/0601745","evidence_quote":"Gives the regularity bound for sums of ideals that motivates the expected inequalities for splitting graphs."},{"cited_title":"Morey, R","cited_arxiv_id":null,"evidence_quote":"Provides the inequality $\\operatorname{bight}(I(G'))-1 \\le \\operatorname{proj\\,dim}(I(G'))$ used to prove projective dimension monotonicity for sequentially Cohen-Macaulay graphs."},{"cited_title":"Katzman, Characteristic-independence of Betti numbers o f graph ideals, J","cited_arxiv_id":null,"evidence_quote":"Provides $\\nu(G')+1 \\le \\operatorname{reg}(I(G'))$, the matching-number lower bound that transfers the regularity inequality to the listed graph families."}],"review_version":1}