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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)."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","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."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","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."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"A connected W_2 graph is 2-quasi-regularizable exactly when n(G) is at least 3α(G)."}],"snapshot_sha256":"fded09d216420119537d7fabc7977a262b859a20498b6ba2ec3eb19f612de38d"},"formal_canon":{"evidence_count":1,"snapshot_sha256":"e407366cb07fa2b72078e62cfb3f762b69ed126331c3f6d4e5b9f6af2be7b05b"},"paper":{"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","authors_text":"Kevin Pereyra","cross_cats":[],"headline":"A connected W_2 graph is 2-quasi-regularizable exactly when n(G) is at least 3α(G).","license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-13T19:55:00Z","title":"The 2-Quasi-Regularizability Conjecture and Independence Polynomials of Wp Graphs"},"references":{"count":37,"internal_anchors":2,"resolved_work":37,"sample":[{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":1,"title":"Y. Alavi, P. J. Malde, A. J. Schwenk, and P. 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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","year":2007}],"snapshot_sha256":"1ff6c16161bfdbd85956682e065f27cafa54a92b5897af3fae23862ea64dbd3e"},"source":{"id":"2605.14076","kind":"arxiv","version":1},"verdict":{"created_at":"2026-05-15T02:33:53.185431Z","id":"f5618d56-e673-460c-805d-cdb4a98f8e19","model_set":{"reader":"grok-4.3"},"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","pith_extraction_headline":"A connected W_2 graph is 2-quasi-regularizable exactly when n(G) is at least 3α(G).","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).","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."}},"verdict_id":"f5618d56-e673-460c-805d-cdb4a98f8e19"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:742c21d1171ede2fa168d4df823db5cd0ec9ff0ea57aa8510f59965572288b7b","target":"record","created_at":"2026-05-17T23:39:12Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"e5e5fda998c68d7c7b1af1b503f218b5100ac8a92839827fad28fc04c3f4362c","cross_cats_sorted":[],"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-13T19:55:00Z","title_canon_sha256":"0fcd5d1f0397675ea15b63c2889b612f4ddbeb9f39dd53ca231aaab28455f5b3"},"schema_version":"1.0","source":{"id":"2605.14076","kind":"arxiv","version":1}},"canonical_sha256":"872b64d94ec4d59679ab4feb746eae40c0dc283ab7cde9ea08de03cdf7ba1f22","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"872b64d94ec4d59679ab4feb746eae40c0dc283ab7cde9ea08de03cdf7ba1f22","first_computed_at":"2026-05-17T23:39:12.362638Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:39:12.362638Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"MdekKzWNL9rmfHuZEd/QZ3TGwO2jzmZQ2OME5MYZjF7M156wRKlx8NJMDF3h7mvShzWOWU3NOx0Y0rCyeakuBw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:39:12.363451Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.14076","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:742c21d1171ede2fa168d4df823db5cd0ec9ff0ea57aa8510f59965572288b7b","sha256:6639b8b313709bd2ac71c0d4a6a18c22db38d9c0c38d60fd9ca67052f830e9c4"],"state_sha256":"9d10cc10485d00d29f1d77016fbb36bb82f6a702e6ef444c3199127efa443b85"}