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We show that the degenerate degree sequence of any graph can be determined by an efficient algorithm. A $k$-independent set in $G$ is any set $S$ of vertices such that $\\Delta(G[S])\\leq k$. The largest cardinality of any $k$-independent set is denoted by $\\alpha_k(G)$. For $k\\in \\{1, 2, 3\\}$, we prove that $\\alpha_{k-1}(G)\\geq {\\sum}_{v\\in G} \\min \\{1, 1/(\\zeta(v)+("},"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":"1507.07194","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-07-26T12:32:34Z","cross_cats_sorted":[],"title_canon_sha256":"84e92fe9963036964244602377887e6e2985bd75ad4243d14289aa34cf34f500","abstract_canon_sha256":"1f5e792224495103a815a46db6d5ecb51e9611328529227f15d61b8a7b8b1413"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:36:16.957183Z","signature_b64":"6ecaC+rDmpIdEZwN7YgTkFmnkNH3mqwEu7g+/38DLQ2/mPZN8PnT09JLM1Fqb2YA/8jl5XHqNo7JZmut8fWlCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e29139ace3b908e48b7c9743a9e0d8534ebabca0ecda0b3acdb2a8471a8f4543","last_reissued_at":"2026-05-18T01:36:16.956652Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:36:16.956652Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Lower bounds for independence and $k$-independence number of graphs using the concept of degenerate degrees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Manouchehr Zaker","submitted_at":"2015-07-26T12:32:34Z","abstract_excerpt":"Let $G$ be a graph and $v$ any vertex of $G$. We define the degenerate degree of $v$, denoted by $\\zeta(v)$ as $\\zeta(v)={\\max}_{H: v\\in H}~\\delta(H)$, where the maximum is taken over all subgraphs of $G$ containing the vertex $v$. We show that the degenerate degree sequence of any graph can be determined by an efficient algorithm. A $k$-independent set in $G$ is any set $S$ of vertices such that $\\Delta(G[S])\\leq k$. The largest cardinality of any $k$-independent set is denoted by $\\alpha_k(G)$. For $k\\in \\{1, 2, 3\\}$, we prove that $\\alpha_{k-1}(G)\\geq {\\sum}_{v\\in G} \\min \\{1, 1/(\\zeta(v)+("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.07194","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":"1507.07194","created_at":"2026-05-18T01:36:16.956741+00:00"},{"alias_kind":"arxiv_version","alias_value":"1507.07194v1","created_at":"2026-05-18T01:36:16.956741+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.07194","created_at":"2026-05-18T01:36:16.956741+00:00"},{"alias_kind":"pith_short_12","alias_value":"4KITTLHDXEEO","created_at":"2026-05-18T12:29:05.191682+00:00"},{"alias_kind":"pith_short_16","alias_value":"4KITTLHDXEEOJC34","created_at":"2026-05-18T12:29:05.191682+00:00"},{"alias_kind":"pith_short_8","alias_value":"4KITTLHD","created_at":"2026-05-18T12:29:05.191682+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/4KITTLHDXEEOJC34S5B2TYGYKN","json":"https://pith.science/pith/4KITTLHDXEEOJC34S5B2TYGYKN.json","graph_json":"https://pith.science/api/pith-number/4KITTLHDXEEOJC34S5B2TYGYKN/graph.json","events_json":"https://pith.science/api/pith-number/4KITTLHDXEEOJC34S5B2TYGYKN/events.json","paper":"https://pith.science/paper/4KITTLHD"},"agent_actions":{"view_html":"https://pith.science/pith/4KITTLHDXEEOJC34S5B2TYGYKN","download_json":"https://pith.science/pith/4KITTLHDXEEOJC34S5B2TYGYKN.json","view_paper":"https://pith.science/paper/4KITTLHD","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1507.07194&json=true","fetch_graph":"https://pith.science/api/pith-number/4KITTLHDXEEOJC34S5B2TYGYKN/graph.json","fetch_events":"https://pith.science/api/pith-number/4KITTLHDXEEOJC34S5B2TYGYKN/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/4KITTLHDXEEOJC34S5B2TYGYKN/action/timestamp_anchor","attest_storage":"https://pith.science/pith/4KITTLHDXEEOJC34S5B2TYGYKN/action/storage_attestation","attest_author":"https://pith.science/pith/4KITTLHDXEEOJC34S5B2TYGYKN/action/author_attestation","sign_citation":"https://pith.science/pith/4KITTLHDXEEOJC34S5B2TYGYKN/action/citation_signature","submit_replication":"https://pith.science/pith/4KITTLHDXEEOJC34S5B2TYGYKN/action/replication_record"}},"created_at":"2026-05-18T01:36:16.956741+00:00","updated_at":"2026-05-18T01:36:16.956741+00:00"}