{"paper":{"title":"The 2-Quasi-Regularizability Conjecture and Independence Polynomials of Wp Graphs","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"A connected W_2 graph is 2-quasi-regularizable exactly when n(G) is at least 3α(G).","cross_cats":[],"primary_cat":"math.CO","authors_text":"Kevin Pereyra","submitted_at":"2026-05-13T19:55:00Z","abstract_excerpt":"Hoang, Levit, Mandrescu and Pham asked for structural conditions ensuring that the independence polynomial of a $\\W_p$ graph is log-concave, or at least unimodal, and conjectured that a connected $\\W_2$ graph is $2$-quasi-regularizable if and only if $n(G)\\ge 3\\alpha(G)$ (2026). We prove the conjecture. The key point is a local expansion theorem: if $G$ is connected and belongs to $\\W_2$, then every non-maximum independent set $A$ satisfies \\[ |N_G(A)|\\ge 2|A|. \\] Thus the only possible obstruction to $2$-quasi-regularizability in a connected $\\W_2$ graph comes from maximum independent sets, w"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We prove the conjecture. The key point is a local expansion theorem: if G is connected and belongs to W_2, then every non-maximum independent set A satisfies |N_G(A)| ≥ 2|A|. Thus the only possible obstruction to 2-quasi-regularizability in a connected W_2 graph comes from maximum independent sets, where the condition is exactly n(G)−α(G)≥2α(G).","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The assumption that G is connected and lies in the class W_2; the local expansion theorem is stated only under these hypotheses, so the reduction to the numerical condition on maximum independent sets holds only inside this restricted family.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Proves the 2-quasi-regularizability conjecture for connected W_2 graphs via a local expansion theorem and derives explicit log-concavity and unimodality regions for their independence polynomials.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A connected W_2 graph is 2-quasi-regularizable exactly when n(G) is at least 3α(G).","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"fded09d216420119537d7fabc7977a262b859a20498b6ba2ec3eb19f612de38d"},"source":{"id":"2605.14076","kind":"arxiv","version":1},"verdict":{"id":"f5618d56-e673-460c-805d-cdb4a98f8e19","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T02:33:53.185431Z","strongest_claim":"We prove the conjecture. The key point is a local expansion theorem: if G is connected and belongs to W_2, then every non-maximum independent set A satisfies |N_G(A)| ≥ 2|A|. Thus the only possible obstruction to 2-quasi-regularizability in a connected W_2 graph comes from maximum independent sets, where the condition is exactly n(G)−α(G)≥2α(G).","one_line_summary":"Proves the 2-quasi-regularizability conjecture for connected W_2 graphs via a local expansion theorem and derives explicit log-concavity and unimodality regions for their independence polynomials.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The assumption that G is connected and lies in the class W_2; the local expansion theorem is stated only under these hypotheses, so the reduction to the numerical condition on maximum independent sets holds only inside this restricted family.","pith_extraction_headline":"A connected W_2 graph is 2-quasi-regularizable exactly when n(G) is at least 3α(G)."},"references":{"count":37,"sample":[{"doi":"","year":1987,"title":"Y. Alavi, P. J. Malde, A. J. Schwenk, and P. Erdos,The vertex independence sequence of a graph is not constrained, Congressus Numerantium58(1987), 15–23","work_id":"a3149759-9ff0-42cf-b3dd-2ec84944bdfa","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1982,"title":"Berge,Some common properties for regularizable graphs, edge-critical graphs and B-graphs, Annals of Discrete Mathematics12(1982), 31–44","work_id":"1ff9ae94-c6d7-4cba-a179-c99bb16d63f7","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2004,"title":"J. I. Brown, C. A. Hickman, and R. J. Nowakowski,On the location of roots of independence polynomials, Journal of Algebraic Combinatorics19(2004), 273–282","work_id":"c697f66b-865f-4554-88d6-724f68067d3b","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2010,"title":"S.-Y. Chen and H.-J. Wang,Unimodality of very well-covered graphs, Ars Combina- toria97A(2010), 509–529","work_id":"e0599989-2042-428e-9c33-f0689480f2db","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2007,"title":"M. Chudnovsky and P. Seymour,The roots of the independence polynomial of a claw-free graph, Journal of Combinatorial Theory, Series B97(2007), 350–357","work_id":"2732c7ce-7806-4c45-b785-124da29f9fa3","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":37,"snapshot_sha256":"1ff6c16161bfdbd85956682e065f27cafa54a92b5897af3fae23862ea64dbd3e","internal_anchors":2},"formal_canon":{"evidence_count":1,"snapshot_sha256":"e407366cb07fa2b72078e62cfb3f762b69ed126331c3f6d4e5b9f6af2be7b05b"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}