{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:NYJEIEXDMZNWLHI3SOHDRFJWAQ","short_pith_number":"pith:NYJEIEXD","schema_version":"1.0","canonical_sha256":"6e124412e3665b659d1b938e3895360433436bd4baa35fd6aab53d54194d52ce","source":{"kind":"arxiv","id":"1605.01455","version":1},"attestation_state":"computed","paper":{"title":"Connectivity Functions and Polymatroids","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Geoff Whittle, Songbao Mo, Susan Jowett","submitted_at":"2016-05-04T23:11:12Z","abstract_excerpt":"A {\\em connectivity function on} a set $E$ is a function $\\lambda:2^E\\rightarrow \\mathbb R$ such that $\\lambda(\\emptyset)=0$, that $\\lambda(X)=\\lambda(E-X)$ for all $X\\subseteq E$ and that $\\lambda(X\\cap Y)+\\lambda(X\\cup Y)\\leq \\lambda(X)+\\lambda(Y)$ for all $X,Y \\subseteq E$. Graphs, matroids and, more generally, polymatroids have associated connectivity functions. We introduce a notion of duality for polymatroids and prove that every connectivity function is the connectivity function of a self-dual polymatroid. We also prove that every integral connectivity function is the connectivity funct"},"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":"1605.01455","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-05-04T23:11:12Z","cross_cats_sorted":[],"title_canon_sha256":"274c74f642150c678b3d9686f0443f4a5a1ee4905896d1442b2e6e436dc308e2","abstract_canon_sha256":"741440d9ae803a6c61a845c64af6d45314571fd4334f8d0a0223d016b035ac28"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:15:33.660295Z","signature_b64":"JsLwHskTpnSYDW43MixeKcbvJuBAe4F240u8MHM6Gu0M15Hr5mXtTbjS+UjdxBQbXWZbbOKxlQD89sQytQL4Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6e124412e3665b659d1b938e3895360433436bd4baa35fd6aab53d54194d52ce","last_reissued_at":"2026-05-18T01:15:33.659659Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:15:33.659659Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Connectivity Functions and Polymatroids","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Geoff Whittle, Songbao Mo, Susan Jowett","submitted_at":"2016-05-04T23:11:12Z","abstract_excerpt":"A {\\em connectivity function on} a set $E$ is a function $\\lambda:2^E\\rightarrow \\mathbb R$ such that $\\lambda(\\emptyset)=0$, that $\\lambda(X)=\\lambda(E-X)$ for all $X\\subseteq E$ and that $\\lambda(X\\cap Y)+\\lambda(X\\cup Y)\\leq \\lambda(X)+\\lambda(Y)$ for all $X,Y \\subseteq E$. Graphs, matroids and, more generally, polymatroids have associated connectivity functions. We introduce a notion of duality for polymatroids and prove that every connectivity function is the connectivity function of a self-dual polymatroid. We also prove that every integral connectivity function is the connectivity funct"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.01455","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":"1605.01455","created_at":"2026-05-18T01:15:33.659736+00:00"},{"alias_kind":"arxiv_version","alias_value":"1605.01455v1","created_at":"2026-05-18T01:15:33.659736+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.01455","created_at":"2026-05-18T01:15:33.659736+00:00"},{"alias_kind":"pith_short_12","alias_value":"NYJEIEXDMZNW","created_at":"2026-05-18T12:30:36.002864+00:00"},{"alias_kind":"pith_short_16","alias_value":"NYJEIEXDMZNWLHI3","created_at":"2026-05-18T12:30:36.002864+00:00"},{"alias_kind":"pith_short_8","alias_value":"NYJEIEXD","created_at":"2026-05-18T12:30:36.002864+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/NYJEIEXDMZNWLHI3SOHDRFJWAQ","json":"https://pith.science/pith/NYJEIEXDMZNWLHI3SOHDRFJWAQ.json","graph_json":"https://pith.science/api/pith-number/NYJEIEXDMZNWLHI3SOHDRFJWAQ/graph.json","events_json":"https://pith.science/api/pith-number/NYJEIEXDMZNWLHI3SOHDRFJWAQ/events.json","paper":"https://pith.science/paper/NYJEIEXD"},"agent_actions":{"view_html":"https://pith.science/pith/NYJEIEXDMZNWLHI3SOHDRFJWAQ","download_json":"https://pith.science/pith/NYJEIEXDMZNWLHI3SOHDRFJWAQ.json","view_paper":"https://pith.science/paper/NYJEIEXD","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1605.01455&json=true","fetch_graph":"https://pith.science/api/pith-number/NYJEIEXDMZNWLHI3SOHDRFJWAQ/graph.json","fetch_events":"https://pith.science/api/pith-number/NYJEIEXDMZNWLHI3SOHDRFJWAQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/NYJEIEXDMZNWLHI3SOHDRFJWAQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/NYJEIEXDMZNWLHI3SOHDRFJWAQ/action/storage_attestation","attest_author":"https://pith.science/pith/NYJEIEXDMZNWLHI3SOHDRFJWAQ/action/author_attestation","sign_citation":"https://pith.science/pith/NYJEIEXDMZNWLHI3SOHDRFJWAQ/action/citation_signature","submit_replication":"https://pith.science/pith/NYJEIEXDMZNWLHI3SOHDRFJWAQ/action/replication_record"}},"created_at":"2026-05-18T01:15:33.659736+00:00","updated_at":"2026-05-18T01:15:33.659736+00:00"}