{"paper":{"title":"A Ridge-Saturation Characterization of $\\alpha$-Critical $\\mathbf {W}_p$ Graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Graphs that are α-critical and in W_p have three equivalent characterizations in graph, complex, and complement terms.","cross_cats":[],"primary_cat":"math.CO","authors_text":"Do Trong Hoang, Eugen Mandrescu, Kevin Pereyra, Vadim E. Levit","submitted_at":"2026-05-16T06:43:26Z","abstract_excerpt":"We characterize the graphs which are simultaneously $\\alpha$-critical and members of the class $\\mathbf W_p$. The characterization is stated in three equivalent languages. In the graph itself, such a graph is a well-covered graph whose codimension-one localization fibers all have size at least $p$ and whose edges are exactly covered by the cliques induced by those fibers. In the independence complex, it is a pure flag complex in which every ridge has degree at least $p$ and every missing edge is generated by the link of a ridge. In the complement, it is a $K_{r+1}$-saturated graph, where $r=\\a"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We characterize the graphs which are simultaneously α-critical and members of the class W_p. The characterization is stated in three equivalent languages. ... This gives an exact formula for the largest p for which a well-covered graph belongs to W_p.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The three descriptions (graph-theoretic fibers, ridge degrees in the independence complex, and (r-1)-clique codegrees in the complement) are equivalent for α-critical members of W_p, relying on the prior definitions of α-criticality and the class W_p without additional verification steps shown in the abstract.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Multi-language characterization of α-critical W_p graphs with saturation consequences, p-bounds, and sharp examples refuting a recent local sufficient condition outside the triangle-free case.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Graphs that are α-critical and in W_p have three equivalent characterizations in graph, complex, and complement terms.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"39250cdc7c38eb868d6f267b971d072c0671865b82008f3f6eebb52bc8864a89"},"source":{"id":"2605.16838","kind":"arxiv","version":1},"verdict":{"id":"4a50035d-8dc1-4181-b5d8-f1af6a5f5d6e","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T21:00:15.372849Z","strongest_claim":"We characterize the graphs which are simultaneously α-critical and members of the class W_p. The characterization is stated in three equivalent languages. ... This gives an exact formula for the largest p for which a well-covered graph belongs to W_p.","one_line_summary":"Multi-language characterization of α-critical W_p graphs with saturation consequences, p-bounds, and sharp examples refuting a recent local sufficient condition outside the triangle-free case.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The three descriptions (graph-theoretic fibers, ridge degrees in the independence complex, and (r-1)-clique codegrees in the complement) are equivalent for α-critical members of W_p, relying on the prior definitions of α-criticality and the class W_p without additional verification steps shown in the abstract.","pith_extraction_headline":"Graphs that are α-critical and in W_p have three equivalent characterizations in graph, complex, and complement terms."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.16838/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T21:31:19.239110Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T21:11:43.327968Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T18:41:56.324014Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T18:33:26.396350Z","status":"skipped","version":"1.0.0","findings_count":0}],"snapshot_sha256":"973c6e19f7cda82162f465b8515b421ada84cedb6baa74caf33937e9e65d9e19"},"references":{"count":27,"sample":[{"doi":"","year":1974,"title":"B. Andrásfai, P. Erdős, and V. T. Sós, On the connection be tween chro- matic number, maximal clique and minimal degree of a graph, Discrete Mathematics 8 (1974), no. 3, 205–218. 12","work_id":"393bcf54-fdf1-402c-adb9-79e4bcf9383e","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1967,"title":"L. W. Beineke, F. Harary, and M. D. Plummer, On the critica l lines of a graph, Pacific Journal of Mathematics 22 (1967), no. 2, 205–212. 2, 5","work_id":"f8b4384d-1594-44f4-9475-41be01f14980","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1982,"title":"Berge, Some common properties for regularizable grap hs, edge- critical graphs and B-graphs, Annals of Discrete Mathematics 12 (1982), 31–44","work_id":"f99873fe-cae4-4ae1-aa97-52e8af6a17a3","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2012,"title":"S. Bermudo and H. Fernau, Lower bounds on the differential of a graph, Discrete Mathematics 312 (2012), 3236–3250. 2","work_id":"69353944-b906-427b-9bb1-5b53c609cf86","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2016,"title":"I. D. Castrillón, R. Cruz, and E. Reyes, On well-covered, vertex decom- posable and Cohen–Macaulay graphs, Electronic Journal of Combina- torics 23 (2016), no. 2, 17 pp. 2","work_id":"c8ecf19f-8a3f-4087-8470-6e7440045c6c","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":27,"snapshot_sha256":"ec530c1f066dc0b922fc389c415423c1eda202054be2b83510b6b4ed75408fb6","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"eb79c9a17414eecb3632ad6f9687e1fb8def25f59ae6ab9fe0db683ddb93cf97"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}