{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:Q4VWJWKOYTKZM6NLJ7VXI3VOID","short_pith_number":"pith:Q4VWJWKO","schema_version":"1.0","canonical_sha256":"872b64d94ec4d59679ab4feb746eae40c0dc283ab7cde9ea08de03cdf7ba1f22","source":{"kind":"arxiv","id":"2605.14076","version":1},"attestation_state":"computed","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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":true,"formal_links_present":true},"canonical_record":{"source":{"id":"2605.14076","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-13T19:55:00Z","cross_cats_sorted":[],"title_canon_sha256":"0fcd5d1f0397675ea15b63c2889b612f4ddbeb9f39dd53ca231aaab28455f5b3","abstract_canon_sha256":"e5e5fda998c68d7c7b1af1b503f218b5100ac8a92839827fad28fc04c3f4362c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:39:12.363451Z","signature_b64":"MdekKzWNL9rmfHuZEd/QZ3TGwO2jzmZQ2OME5MYZjF7M156wRKlx8NJMDF3h7mvShzWOWU3NOx0Y0rCyeakuBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"872b64d94ec4d59679ab4feb746eae40c0dc283ab7cde9ea08de03cdf7ba1f22","last_reissued_at":"2026-05-17T23:39:12.362638Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:39:12.362638Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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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. 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