{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:OBVZVTFOOWCI6GY4YM5LJOQJ5X","short_pith_number":"pith:OBVZVTFO","schema_version":"1.0","canonical_sha256":"706b9accae75848f1b1cc33ab4ba09edde715d97ac1502316824dfd7c3bb8a22","source":{"kind":"arxiv","id":"2606.22972","version":1},"attestation_state":"computed","paper":{"title":"On externally supported independence number of graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Adriana Roux, Aleksandra Tepeh, Dragana Bo\\v{z}ovi\\'c, Iztok Peterin","submitted_at":"2026-06-22T07:52:43Z","abstract_excerpt":"We introduce the \\emph{externally supported independence number} $\\alpha_{\\rm es}(G)$ of a graph $G$ as the maximum cardinality of an independent set $B$ with an additional condition, that vertices from $N(B)$ are dominated by vertices in $V(G)-N[B]$. This parameter yields an improved upper bound on the isolation number $\\iota(G)$. We show that computing $\\alpha_{\\rm es}(G)$ is NP-hard, while for trees we present a linear-time algorithm. We also establish several sharp bounds on $\\alpha_{\\rm es}(G)$ for general graphs, with additional refined results for trees. In several cases, we completely "},"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":"2606.22972","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-22T07:52:43Z","cross_cats_sorted":[],"title_canon_sha256":"be50487abe7e8dbab427c1071790d2f9c896b71988be6371e4e8a814c410fb03","abstract_canon_sha256":"2cdeabc3cded5266d81ed6f77401527730ccc873a789b6771fb2b9a85d891c46"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-23T03:14:05.630592Z","signature_b64":"cr6grOcak00/yk2TM4W6GG9s/q+L3ybJOqlFtDhned8n/yJXzk+QW03Hsxnv0iPfV8ES/2geZ2M6168g4S78AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"706b9accae75848f1b1cc33ab4ba09edde715d97ac1502316824dfd7c3bb8a22","last_reissued_at":"2026-06-23T03:14:05.630224Z","signature_status":"signed_v1","first_computed_at":"2026-06-23T03:14:05.630224Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On externally supported independence number of graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Adriana Roux, Aleksandra Tepeh, Dragana Bo\\v{z}ovi\\'c, Iztok Peterin","submitted_at":"2026-06-22T07:52:43Z","abstract_excerpt":"We introduce the \\emph{externally supported independence number} $\\alpha_{\\rm es}(G)$ of a graph $G$ as the maximum cardinality of an independent set $B$ with an additional condition, that vertices from $N(B)$ are dominated by vertices in $V(G)-N[B]$. This parameter yields an improved upper bound on the isolation number $\\iota(G)$. We show that computing $\\alpha_{\\rm es}(G)$ is NP-hard, while for trees we present a linear-time algorithm. We also establish several sharp bounds on $\\alpha_{\\rm es}(G)$ for general graphs, with additional refined results for trees. In several cases, we completely "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.22972","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/2606.22972/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":"2606.22972","created_at":"2026-06-23T03:14:05.630283+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.22972v1","created_at":"2026-06-23T03:14:05.630283+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.22972","created_at":"2026-06-23T03:14:05.630283+00:00"},{"alias_kind":"pith_short_12","alias_value":"OBVZVTFOOWCI","created_at":"2026-06-23T03:14:05.630283+00:00"},{"alias_kind":"pith_short_16","alias_value":"OBVZVTFOOWCI6GY4","created_at":"2026-06-23T03:14:05.630283+00:00"},{"alias_kind":"pith_short_8","alias_value":"OBVZVTFO","created_at":"2026-06-23T03:14:05.630283+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/OBVZVTFOOWCI6GY4YM5LJOQJ5X","json":"https://pith.science/pith/OBVZVTFOOWCI6GY4YM5LJOQJ5X.json","graph_json":"https://pith.science/api/pith-number/OBVZVTFOOWCI6GY4YM5LJOQJ5X/graph.json","events_json":"https://pith.science/api/pith-number/OBVZVTFOOWCI6GY4YM5LJOQJ5X/events.json","paper":"https://pith.science/paper/OBVZVTFO"},"agent_actions":{"view_html":"https://pith.science/pith/OBVZVTFOOWCI6GY4YM5LJOQJ5X","download_json":"https://pith.science/pith/OBVZVTFOOWCI6GY4YM5LJOQJ5X.json","view_paper":"https://pith.science/paper/OBVZVTFO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.22972&json=true","fetch_graph":"https://pith.science/api/pith-number/OBVZVTFOOWCI6GY4YM5LJOQJ5X/graph.json","fetch_events":"https://pith.science/api/pith-number/OBVZVTFOOWCI6GY4YM5LJOQJ5X/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/OBVZVTFOOWCI6GY4YM5LJOQJ5X/action/timestamp_anchor","attest_storage":"https://pith.science/pith/OBVZVTFOOWCI6GY4YM5LJOQJ5X/action/storage_attestation","attest_author":"https://pith.science/pith/OBVZVTFOOWCI6GY4YM5LJOQJ5X/action/author_attestation","sign_citation":"https://pith.science/pith/OBVZVTFOOWCI6GY4YM5LJOQJ5X/action/citation_signature","submit_replication":"https://pith.science/pith/OBVZVTFOOWCI6GY4YM5LJOQJ5X/action/replication_record"}},"created_at":"2026-06-23T03:14:05.630283+00:00","updated_at":"2026-06-23T03:14:05.630283+00:00"}