{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:GBN7AEEAKM4W7GFQCAOXRU5HUG","short_pith_number":"pith:GBN7AEEA","schema_version":"1.0","canonical_sha256":"305bf0108053396f98b0101d78d3a7a19e69a4840cdead6ca86a7cd8079f6e13","source":{"kind":"arxiv","id":"1601.02099","version":2},"attestation_state":"computed","paper":{"title":"Dynamic Monopolies for Degree Proportional Thresholds in Connected Graphs of Girth at least Five and Trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dieter Rautenbach, Michael Gentner","submitted_at":"2016-01-09T12:06:15Z","abstract_excerpt":"Let $G$ be a graph, and let $\\rho\\in (0,1)$. For a set $D$ of vertices of $G$, let the set $H_{\\rho}(D)$ arise by starting with the set $D$, and iteratively adding further vertices $u$ to the current set if they have at least $\\lceil \\rho d_G(u)\\rceil$ neighbors in it. If $H_{\\rho}(D)$ contains all vertices of $G$, then $D$ is known as an irreversible dynamic monopoly or a perfect target set associated with the threshold function $u\\mapsto \\lceil \\rho d_G(u)\\rceil$. Let $h_{\\rho}(G)$ be the minimum cardinality of such an irreversible dynamic monopoly.\n  For a connected graph $G$ of maximum deg"},"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":"1601.02099","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-01-09T12:06:15Z","cross_cats_sorted":[],"title_canon_sha256":"d0f662be83e1d6853d5b9b9b267b2df54c68ce02ffeec1ef760f674424d66cb5","abstract_canon_sha256":"633f00822a788cf01540c1c38130ac690d36d98b5ac09ab1ae4b345a6141b4a7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:20:50.066296Z","signature_b64":"8BUE3wD1fF/kIogQVChCAgce0ci/nqm6qK5mNlPcZYXMOx5ZqCtnopNSthhTVam6NnLKDr3cyuNH2sCTWwclCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"305bf0108053396f98b0101d78d3a7a19e69a4840cdead6ca86a7cd8079f6e13","last_reissued_at":"2026-05-18T01:20:50.065815Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:20:50.065815Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Dynamic Monopolies for Degree Proportional Thresholds in Connected Graphs of Girth at least Five and Trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dieter Rautenbach, Michael Gentner","submitted_at":"2016-01-09T12:06:15Z","abstract_excerpt":"Let $G$ be a graph, and let $\\rho\\in (0,1)$. For a set $D$ of vertices of $G$, let the set $H_{\\rho}(D)$ arise by starting with the set $D$, and iteratively adding further vertices $u$ to the current set if they have at least $\\lceil \\rho d_G(u)\\rceil$ neighbors in it. If $H_{\\rho}(D)$ contains all vertices of $G$, then $D$ is known as an irreversible dynamic monopoly or a perfect target set associated with the threshold function $u\\mapsto \\lceil \\rho d_G(u)\\rceil$. Let $h_{\\rho}(G)$ be the minimum cardinality of such an irreversible dynamic monopoly.\n  For a connected graph $G$ of maximum deg"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.02099","kind":"arxiv","version":2},"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":"1601.02099","created_at":"2026-05-18T01:20:50.065884+00:00"},{"alias_kind":"arxiv_version","alias_value":"1601.02099v2","created_at":"2026-05-18T01:20:50.065884+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.02099","created_at":"2026-05-18T01:20:50.065884+00:00"},{"alias_kind":"pith_short_12","alias_value":"GBN7AEEAKM4W","created_at":"2026-05-18T12:30:15.759754+00:00"},{"alias_kind":"pith_short_16","alias_value":"GBN7AEEAKM4W7GFQ","created_at":"2026-05-18T12:30:15.759754+00:00"},{"alias_kind":"pith_short_8","alias_value":"GBN7AEEA","created_at":"2026-05-18T12:30:15.759754+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/GBN7AEEAKM4W7GFQCAOXRU5HUG","json":"https://pith.science/pith/GBN7AEEAKM4W7GFQCAOXRU5HUG.json","graph_json":"https://pith.science/api/pith-number/GBN7AEEAKM4W7GFQCAOXRU5HUG/graph.json","events_json":"https://pith.science/api/pith-number/GBN7AEEAKM4W7GFQCAOXRU5HUG/events.json","paper":"https://pith.science/paper/GBN7AEEA"},"agent_actions":{"view_html":"https://pith.science/pith/GBN7AEEAKM4W7GFQCAOXRU5HUG","download_json":"https://pith.science/pith/GBN7AEEAKM4W7GFQCAOXRU5HUG.json","view_paper":"https://pith.science/paper/GBN7AEEA","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1601.02099&json=true","fetch_graph":"https://pith.science/api/pith-number/GBN7AEEAKM4W7GFQCAOXRU5HUG/graph.json","fetch_events":"https://pith.science/api/pith-number/GBN7AEEAKM4W7GFQCAOXRU5HUG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GBN7AEEAKM4W7GFQCAOXRU5HUG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GBN7AEEAKM4W7GFQCAOXRU5HUG/action/storage_attestation","attest_author":"https://pith.science/pith/GBN7AEEAKM4W7GFQCAOXRU5HUG/action/author_attestation","sign_citation":"https://pith.science/pith/GBN7AEEAKM4W7GFQCAOXRU5HUG/action/citation_signature","submit_replication":"https://pith.science/pith/GBN7AEEAKM4W7GFQCAOXRU5HUG/action/replication_record"}},"created_at":"2026-05-18T01:20:50.065884+00:00","updated_at":"2026-05-18T01:20:50.065884+00:00"}