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Let $\\Omega(G)$ denote the family of all maximum independent sets. A collection $F$ of sets is an hke collection if $|\\bigcup \\Gamma|+|\\bigcap \\Gamma|=2\\alpha$ holds for every subcollection $\\Gamma$ of $F$. We characterize an hke collection and invoke "},"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":"1603.06552","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-03-21T19:36:16Z","cross_cats_sorted":[],"title_canon_sha256":"0e40c1dc2e9b66bceb437ae12e3f315ef58fb45d3ff8e20d6cf2e4253d0ab828","abstract_canon_sha256":"7748205f03a8325ac19c1b12296963db8d5ff10c6f288844ad686245f4c7fa79"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:18:13.657151Z","signature_b64":"8eSQIYfTOVyJQvVHMN/ObISFcH1ld1mIPz8PzqeAoN+DJyrQcsCzG9Go2FlpJhFHSXvquEwgnxFAD/gZzpSlCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"40fc9e49be88ad24e46f8e97f1cadac14b5013159235c25c363ea2e52b4b9e84","last_reissued_at":"2026-05-18T01:18:13.656429Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:18:13.656429Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Hereditary Konig Egervary Collections","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Adi Jarden","submitted_at":"2016-03-21T19:36:16Z","abstract_excerpt":"Let $G$ be a simple graph with vertex set $V(G)$. A subset $S$ of $V(G)$ is independent if no two vertices from $S$ are adjacent. The graph $G$ is known to be a Konig-Egervary (KE in short) graph if $\\alpha(G) + \\mu(G)= |V(G)|$, where $\\alpha(G)$ denotes the size of a maximum independent set and $\\mu(G)$ is the cardinality of a maximum matching. Let $\\Omega(G)$ denote the family of all maximum independent sets. A collection $F$ of sets is an hke collection if $|\\bigcup \\Gamma|+|\\bigcap \\Gamma|=2\\alpha$ holds for every subcollection $\\Gamma$ of $F$. We characterize an hke collection and invoke "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.06552","kind":"arxiv","version":3},"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":"1603.06552","created_at":"2026-05-18T01:18:13.656545+00:00"},{"alias_kind":"arxiv_version","alias_value":"1603.06552v3","created_at":"2026-05-18T01:18:13.656545+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.06552","created_at":"2026-05-18T01:18:13.656545+00:00"},{"alias_kind":"pith_short_12","alias_value":"ID6J4SN6RCWS","created_at":"2026-05-18T12:30:22.444734+00:00"},{"alias_kind":"pith_short_16","alias_value":"ID6J4SN6RCWSJZDP","created_at":"2026-05-18T12:30:22.444734+00:00"},{"alias_kind":"pith_short_8","alias_value":"ID6J4SN6","created_at":"2026-05-18T12:30:22.444734+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/ID6J4SN6RCWSJZDPR2L7DSW2YF","json":"https://pith.science/pith/ID6J4SN6RCWSJZDPR2L7DSW2YF.json","graph_json":"https://pith.science/api/pith-number/ID6J4SN6RCWSJZDPR2L7DSW2YF/graph.json","events_json":"https://pith.science/api/pith-number/ID6J4SN6RCWSJZDPR2L7DSW2YF/events.json","paper":"https://pith.science/paper/ID6J4SN6"},"agent_actions":{"view_html":"https://pith.science/pith/ID6J4SN6RCWSJZDPR2L7DSW2YF","download_json":"https://pith.science/pith/ID6J4SN6RCWSJZDPR2L7DSW2YF.json","view_paper":"https://pith.science/paper/ID6J4SN6","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1603.06552&json=true","fetch_graph":"https://pith.science/api/pith-number/ID6J4SN6RCWSJZDPR2L7DSW2YF/graph.json","fetch_events":"https://pith.science/api/pith-number/ID6J4SN6RCWSJZDPR2L7DSW2YF/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ID6J4SN6RCWSJZDPR2L7DSW2YF/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ID6J4SN6RCWSJZDPR2L7DSW2YF/action/storage_attestation","attest_author":"https://pith.science/pith/ID6J4SN6RCWSJZDPR2L7DSW2YF/action/author_attestation","sign_citation":"https://pith.science/pith/ID6J4SN6RCWSJZDPR2L7DSW2YF/action/citation_signature","submit_replication":"https://pith.science/pith/ID6J4SN6RCWSJZDPR2L7DSW2YF/action/replication_record"}},"created_at":"2026-05-18T01:18:13.656545+00:00","updated_at":"2026-05-18T01:18:13.656545+00:00"}