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A set $S\\subseteq V\\left( G\\right) $ is independent if no two vertices from $S$ are adjacent, and by $\\mathrm{Ind}(G)$ we mean the family of all independent sets of $G$.\n  The number $d\\left( X\\right) =$ $\\left\\vert X\\right\\vert -\\left\\vert N(X)\\right\\vert $ is the difference of $X\\subseteq V\\left( G\\right) $, and a set $A\\in\\mathrm{Ind}(G)$ is critical if $d(A)=\\max \\{d\\left( I\\right) :I\\in\\mathrm{Ind}(G)\\}$ (Zhang, 1990).\n  Let us recall the following definitions:\n  $\\mathrm{core}\\left( G\\right) $ = $\\bigcap$ {S : S is a maximum "},"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":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1407.7368","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2014-07-28T09:36:29Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"eebdc111dd130d5316cff18de0dd4c10d3af0a1bc64594cc0290f1d2007fd353","abstract_canon_sha256":"d9693ee7e7417aef8cd6cc968b34a729e5392d33ae82c47a9d245f7cce1ab613"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:46:23.290072Z","signature_b64":"NEAaDZ/6Oun0cO/SqhkxHMD5kN5vk4rF+nD2nWgpzq6mcekDTSvACrUm9WwU/hhHbuJMNByJXJc0flnmpXXEBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2fbe71e9a72ca28396e16f61ddb7601058461e9f93b92be1d1941663a1f9b435","last_reissued_at":"2026-05-18T02:46:23.289671Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:46:23.289671Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Critical Independent Sets of a Graph","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.DM","authors_text":"Eugen Mandrescu, Vadim E. 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A set $S\\subseteq V\\left( G\\right) $ is independent if no two vertices from $S$ are adjacent, and by $\\mathrm{Ind}(G)$ we mean the family of all independent sets of $G$.\n  The number $d\\left( X\\right) =$ $\\left\\vert X\\right\\vert -\\left\\vert N(X)\\right\\vert $ is the difference of $X\\subseteq V\\left( G\\right) $, and a set $A\\in\\mathrm{Ind}(G)$ is critical if $d(A)=\\max \\{d\\left( I\\right) :I\\in\\mathrm{Ind}(G)\\}$ (Zhang, 1990).\n  Let us recall the following definitions:\n  $\\mathrm{core}\\left( G\\right) $ = $\\bigcap$ {S : S is a maximum "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.7368","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1407.7368","created_at":"2026-05-18T02:46:23.289728+00:00"},{"alias_kind":"arxiv_version","alias_value":"1407.7368v1","created_at":"2026-05-18T02:46:23.289728+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1407.7368","created_at":"2026-05-18T02:46:23.289728+00:00"},{"alias_kind":"pith_short_12","alias_value":"F67HD2NHFSRI","created_at":"2026-05-18T12:28:28.263976+00:00"},{"alias_kind":"pith_short_16","alias_value":"F67HD2NHFSRIHFXB","created_at":"2026-05-18T12:28:28.263976+00:00"},{"alias_kind":"pith_short_8","alias_value":"F67HD2NH","created_at":"2026-05-18T12:28:28.263976+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/F67HD2NHFSRIHFXBN5Q53N3ACB","json":"https://pith.science/pith/F67HD2NHFSRIHFXBN5Q53N3ACB.json","graph_json":"https://pith.science/api/pith-number/F67HD2NHFSRIHFXBN5Q53N3ACB/graph.json","events_json":"https://pith.science/api/pith-number/F67HD2NHFSRIHFXBN5Q53N3ACB/events.json","paper":"https://pith.science/paper/F67HD2NH"},"agent_actions":{"view_html":"https://pith.science/pith/F67HD2NHFSRIHFXBN5Q53N3ACB","download_json":"https://pith.science/pith/F67HD2NHFSRIHFXBN5Q53N3ACB.json","view_paper":"https://pith.science/paper/F67HD2NH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1407.7368&json=true","fetch_graph":"https://pith.science/api/pith-number/F67HD2NHFSRIHFXBN5Q53N3ACB/graph.json","fetch_events":"https://pith.science/api/pith-number/F67HD2NHFSRIHFXBN5Q53N3ACB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/F67HD2NHFSRIHFXBN5Q53N3ACB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/F67HD2NHFSRIHFXBN5Q53N3ACB/action/storage_attestation","attest_author":"https://pith.science/pith/F67HD2NHFSRIHFXBN5Q53N3ACB/action/author_attestation","sign_citation":"https://pith.science/pith/F67HD2NHFSRIHFXBN5Q53N3ACB/action/citation_signature","submit_replication":"https://pith.science/pith/F67HD2NHFSRIHFXBN5Q53N3ACB/action/replication_record"}},"created_at":"2026-05-18T02:46:23.289728+00:00","updated_at":"2026-05-18T02:46:23.289728+00:00"}