{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:VWONK2HMCYJ2EHCSUDKPCIYUDS","short_pith_number":"pith:VWONK2HM","schema_version":"1.0","canonical_sha256":"ad9cd568ec1613a21c52a0d4f123141c95443c1e99d365158e28f0329defdefe","source":{"kind":"arxiv","id":"2509.03150","version":1},"attestation_state":"computed","paper":{"title":"Sparsity, Stress-Independence and Globally Linked Pairs in Graph Rigidity Theory","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.CO","authors_text":"Bill Jackson, D\\'aniel Garamv\\\"olgyi, Tibor Jord\\'an","submitted_at":"2025-09-03T09:03:24Z","abstract_excerpt":"A graph is $\\mathcal{R}_d$-independent (resp. $\\mathcal{R}_d$-connected) if its $d$-dimensional generic rigidity matroid is free (resp. connected). A result of Maxwell from 1867 implies that every $\\mathcal{R}_d$-independent graph satisfies the sparsity condition $|E(H)|\\leq d|V(H)|-\\binom{d+1}{2}$ for all subgraphs $H$ with at least $d+1$ vertices. Several other families of graphs $G$ arising naturally in rigidity theory, such as minimally globally $d$-rigid graphs, are known to satisfy the bound $|E(G)|\\leq (d+1)|V(G)|-\\binom{d+2}{2}$. We unify and extend these results by considering the fam"},"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":"2509.03150","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-09-03T09:03:24Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"80625e0603880f0be4a4d1ab95ecd8612ec92e5c165995eb93d2783d9f0a39f3","abstract_canon_sha256":"f0b0b4f31ef1e4941498ae794acc39dd8b538d0c67d7e2002463e81d47b68c6b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T12:04:05.023709Z","signature_b64":"OxaHr2FdfS3mVf378AwF1Ie3ZqGjFHjxDn313HfZ2mpCpaZk30/ZofM+kGDqBPTu06kvRPpcXboZzqeBKf8hAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ad9cd568ec1613a21c52a0d4f123141c95443c1e99d365158e28f0329defdefe","last_reissued_at":"2026-07-05T12:04:05.023265Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T12:04:05.023265Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sparsity, Stress-Independence and Globally Linked Pairs in Graph Rigidity Theory","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.CO","authors_text":"Bill Jackson, D\\'aniel Garamv\\\"olgyi, Tibor Jord\\'an","submitted_at":"2025-09-03T09:03:24Z","abstract_excerpt":"A graph is $\\mathcal{R}_d$-independent (resp. $\\mathcal{R}_d$-connected) if its $d$-dimensional generic rigidity matroid is free (resp. connected). A result of Maxwell from 1867 implies that every $\\mathcal{R}_d$-independent graph satisfies the sparsity condition $|E(H)|\\leq d|V(H)|-\\binom{d+1}{2}$ for all subgraphs $H$ with at least $d+1$ vertices. Several other families of graphs $G$ arising naturally in rigidity theory, such as minimally globally $d$-rigid graphs, are known to satisfy the bound $|E(G)|\\leq (d+1)|V(G)|-\\binom{d+2}{2}$. We unify and extend these results by considering the fam"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.03150","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2509.03150/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2509.03150","created_at":"2026-07-05T12:04:05.023323+00:00"},{"alias_kind":"arxiv_version","alias_value":"2509.03150v1","created_at":"2026-07-05T12:04:05.023323+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2509.03150","created_at":"2026-07-05T12:04:05.023323+00:00"},{"alias_kind":"pith_short_12","alias_value":"VWONK2HMCYJ2","created_at":"2026-07-05T12:04:05.023323+00:00"},{"alias_kind":"pith_short_16","alias_value":"VWONK2HMCYJ2EHCS","created_at":"2026-07-05T12:04:05.023323+00:00"},{"alias_kind":"pith_short_8","alias_value":"VWONK2HM","created_at":"2026-07-05T12:04:05.023323+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2604.27989","citing_title":"Cliques in minimally globally rigid graphs","ref_index":3,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/VWONK2HMCYJ2EHCSUDKPCIYUDS","json":"https://pith.science/pith/VWONK2HMCYJ2EHCSUDKPCIYUDS.json","graph_json":"https://pith.science/api/pith-number/VWONK2HMCYJ2EHCSUDKPCIYUDS/graph.json","events_json":"https://pith.science/api/pith-number/VWONK2HMCYJ2EHCSUDKPCIYUDS/events.json","paper":"https://pith.science/paper/VWONK2HM"},"agent_actions":{"view_html":"https://pith.science/pith/VWONK2HMCYJ2EHCSUDKPCIYUDS","download_json":"https://pith.science/pith/VWONK2HMCYJ2EHCSUDKPCIYUDS.json","view_paper":"https://pith.science/paper/VWONK2HM","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2509.03150&json=true","fetch_graph":"https://pith.science/api/pith-number/VWONK2HMCYJ2EHCSUDKPCIYUDS/graph.json","fetch_events":"https://pith.science/api/pith-number/VWONK2HMCYJ2EHCSUDKPCIYUDS/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VWONK2HMCYJ2EHCSUDKPCIYUDS/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VWONK2HMCYJ2EHCSUDKPCIYUDS/action/storage_attestation","attest_author":"https://pith.science/pith/VWONK2HMCYJ2EHCSUDKPCIYUDS/action/author_attestation","sign_citation":"https://pith.science/pith/VWONK2HMCYJ2EHCSUDKPCIYUDS/action/citation_signature","submit_replication":"https://pith.science/pith/VWONK2HMCYJ2EHCSUDKPCIYUDS/action/replication_record"}},"created_at":"2026-07-05T12:04:05.023323+00:00","updated_at":"2026-07-05T12:04:05.023323+00:00"}