{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:C3VK4LFYG4PMDMY44LM3VNFZWX","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"e751e23aad8813b53e87cb4361e701e1f97941718c99835925bf3dad50c3a4d1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-05-23T17:38:31Z","title_canon_sha256":"a716f7f6190582446886d0ab1c07de21c4a13795b4e30feaee7d011fc64dedd3"},"schema_version":"1.0","source":{"id":"1405.6138","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1405.6138","created_at":"2026-05-18T02:51:14Z"},{"alias_kind":"arxiv_version","alias_value":"1405.6138v1","created_at":"2026-05-18T02:51:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1405.6138","created_at":"2026-05-18T02:51:14Z"},{"alias_kind":"pith_short_12","alias_value":"C3VK4LFYG4PM","created_at":"2026-05-18T12:28:22Z"},{"alias_kind":"pith_short_16","alias_value":"C3VK4LFYG4PMDMY4","created_at":"2026-05-18T12:28:22Z"},{"alias_kind":"pith_short_8","alias_value":"C3VK4LFY","created_at":"2026-05-18T12:28:22Z"}],"graph_snapshots":[{"event_id":"sha256:d045a3f1adc742041b8d51d4d91b5d36fde8c46d9a4d193e9a84ebb60d0a5fd4","target":"graph","created_at":"2026-05-18T02:51:14Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Let $G$ be a graph and $\\tau$ be an assignment of nonnegative integer thresholds to the vertices of $G$. A subset of vertices $D$ is said to be a $\\tau$-dynamic monopoly, if $V(G)$ can be partitioned into subsets $D_0, D_1, \\ldots, D_k$ such that $D_0=D$ and for any $i\\in \\{0, \\ldots, k-1\\}$, each vertex $v$ in $D_{i+1}$ has at least $\\tau(v)$ neighbors in $D_0\\cup \\ldots \\cup D_i$. Denote the size of smallest $\\tau$-dynamic monopoly by $dyn_{\\tau}(G)$ and the average of thresholds in $\\tau$ by $\\overline{\\tau}$. We show that the values of $dyn_{\\tau}(G)$ over all assignments $\\tau$ with the s","authors_text":"Kaveh Khoshkhah, Manouchehr Zaker","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-05-23T17:38:31Z","title":"On the largest dynamic monopolies of graphs with a given average threshold"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.6138","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:441026c7f5c24a7febcb853cb5c0afb1044be6f61f6a7f03109bac633dd24f7a","target":"record","created_at":"2026-05-18T02:51:14Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"e751e23aad8813b53e87cb4361e701e1f97941718c99835925bf3dad50c3a4d1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-05-23T17:38:31Z","title_canon_sha256":"a716f7f6190582446886d0ab1c07de21c4a13795b4e30feaee7d011fc64dedd3"},"schema_version":"1.0","source":{"id":"1405.6138","kind":"arxiv","version":1}},"canonical_sha256":"16eaae2cb8371ec1b31ce2d9bab4b9b5e62f8a06e292ab5b06b5f0d0bae109ff","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"16eaae2cb8371ec1b31ce2d9bab4b9b5e62f8a06e292ab5b06b5f0d0bae109ff","first_computed_at":"2026-05-18T02:51:14.011388Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:51:14.011388Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"kZSFlcpqSGqjc7cc6wjs2uI1KQJwe7pRNtvtVL2FZYthAauyAnfJGxBvYCrln6FH21v5ALkNCgpEyV3DGyJ2Bg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:51:14.012105Z","signed_message":"canonical_sha256_bytes"},"source_id":"1405.6138","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:441026c7f5c24a7febcb853cb5c0afb1044be6f61f6a7f03109bac633dd24f7a","sha256:d045a3f1adc742041b8d51d4d91b5d36fde8c46d9a4d193e9a84ebb60d0a5fd4"],"state_sha256":"21a61a7b36dcad36865164119ea59d9546a69b7d0767f8e5183532f4aefdada9"}