{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:FRKIHYNMEG6TOQUXQOWO4OJOPW","short_pith_number":"pith:FRKIHYNM","canonical_record":{"source":{"id":"1110.4310","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-10-19T15:25:43Z","cross_cats_sorted":[],"title_canon_sha256":"aa182e2069a16debc2fafd5113005dfd44fd52dc6334e1af7ddefdf283f0f618","abstract_canon_sha256":"f2f6fee69b79eb91bc46bc75a90bc76dfbeff12355efdc81fa6faec0fc05ace2"},"schema_version":"1.0"},"canonical_sha256":"2c5483e1ac21bd37429783acee392e7da4c573e57c2b9853da5994ef433ac42c","source":{"kind":"arxiv","id":"1110.4310","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1110.4310","created_at":"2026-05-18T04:10:37Z"},{"alias_kind":"arxiv_version","alias_value":"1110.4310v1","created_at":"2026-05-18T04:10:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.4310","created_at":"2026-05-18T04:10:37Z"},{"alias_kind":"pith_short_12","alias_value":"FRKIHYNMEG6T","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_16","alias_value":"FRKIHYNMEG6TOQUX","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_8","alias_value":"FRKIHYNM","created_at":"2026-05-18T12:26:28Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:FRKIHYNMEG6TOQUXQOWO4OJOPW","target":"record","payload":{"canonical_record":{"source":{"id":"1110.4310","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-10-19T15:25:43Z","cross_cats_sorted":[],"title_canon_sha256":"aa182e2069a16debc2fafd5113005dfd44fd52dc6334e1af7ddefdf283f0f618","abstract_canon_sha256":"f2f6fee69b79eb91bc46bc75a90bc76dfbeff12355efdc81fa6faec0fc05ace2"},"schema_version":"1.0"},"canonical_sha256":"2c5483e1ac21bd37429783acee392e7da4c573e57c2b9853da5994ef433ac42c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:10:37.685296Z","signature_b64":"L+nLQko6JxkE3iSo9HG4xkCYcqxygSNIehewtWxdgFXz17oe3tY837l0HsddJh2+U7LYfdFaN36HZrHvQ47iAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2c5483e1ac21bd37429783acee392e7da4c573e57c2b9853da5994ef433ac42c","last_reissued_at":"2026-05-18T04:10:37.684531Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:10:37.684531Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1110.4310","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:10:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6+tMGVsUhMy6rh7ad1kCpSh1SUPC1a0ID3/HcS+jCHcnLaUMfBaztWtleB0bM9S0jYWXU5uvP5zQ5L9FR3oBDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T11:38:41.471205Z"},"content_sha256":"de39fba6e769a582ccd9bd90c5694d2aefc0fc860751fbb121a46d8f63bab71e","schema_version":"1.0","event_id":"sha256:de39fba6e769a582ccd9bd90c5694d2aefc0fc860751fbb121a46d8f63bab71e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:FRKIHYNMEG6TOQUXQOWO4OJOPW","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On graphs having maximal independent sets of exactly $t$ distinct cardinalities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bert L. Hartnell, Douglas F. Rall","submitted_at":"2011-10-19T15:25:43Z","abstract_excerpt":"For a given positive integer t we consider graphs having maximal independent sets of precisely t distinct cardinalities and restrict our attention to those that have no vertices of degree one. In the situation when t is four or larger and the length of the shortest cycle is at least 6t-6, we completely characterize such graphs."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.4310","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":""},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:10:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qMG65rr6R3MqdsJjYEFmpPlzxPjS6I7xz1wk0vwmxtSALH1x41lX/FpDobdt+Ah4r/c+5Hs82b8k6BLnLnf+Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T11:38:41.471568Z"},"content_sha256":"120893ae837987865f3e512601a4e0f3305a32000eaace2090e24ec1777483e4","schema_version":"1.0","event_id":"sha256:120893ae837987865f3e512601a4e0f3305a32000eaace2090e24ec1777483e4"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FRKIHYNMEG6TOQUXQOWO4OJOPW/bundle.json","state_url":"https://pith.science/pith/FRKIHYNMEG6TOQUXQOWO4OJOPW/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FRKIHYNMEG6TOQUXQOWO4OJOPW/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-07T11:38:41Z","links":{"resolver":"https://pith.science/pith/FRKIHYNMEG6TOQUXQOWO4OJOPW","bundle":"https://pith.science/pith/FRKIHYNMEG6TOQUXQOWO4OJOPW/bundle.json","state":"https://pith.science/pith/FRKIHYNMEG6TOQUXQOWO4OJOPW/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FRKIHYNMEG6TOQUXQOWO4OJOPW/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:FRKIHYNMEG6TOQUXQOWO4OJOPW","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":"f2f6fee69b79eb91bc46bc75a90bc76dfbeff12355efdc81fa6faec0fc05ace2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-10-19T15:25:43Z","title_canon_sha256":"aa182e2069a16debc2fafd5113005dfd44fd52dc6334e1af7ddefdf283f0f618"},"schema_version":"1.0","source":{"id":"1110.4310","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1110.4310","created_at":"2026-05-18T04:10:37Z"},{"alias_kind":"arxiv_version","alias_value":"1110.4310v1","created_at":"2026-05-18T04:10:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.4310","created_at":"2026-05-18T04:10:37Z"},{"alias_kind":"pith_short_12","alias_value":"FRKIHYNMEG6T","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_16","alias_value":"FRKIHYNMEG6TOQUX","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_8","alias_value":"FRKIHYNM","created_at":"2026-05-18T12:26:28Z"}],"graph_snapshots":[{"event_id":"sha256:120893ae837987865f3e512601a4e0f3305a32000eaace2090e24ec1777483e4","target":"graph","created_at":"2026-05-18T04:10:37Z","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":"For a given positive integer t we consider graphs having maximal independent sets of precisely t distinct cardinalities and restrict our attention to those that have no vertices of degree one. In the situation when t is four or larger and the length of the shortest cycle is at least 6t-6, we completely characterize such graphs.","authors_text":"Bert L. Hartnell, Douglas F. Rall","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-10-19T15:25:43Z","title":"On graphs having maximal independent sets of exactly $t$ distinct cardinalities"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.4310","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:de39fba6e769a582ccd9bd90c5694d2aefc0fc860751fbb121a46d8f63bab71e","target":"record","created_at":"2026-05-18T04:10:37Z","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":"f2f6fee69b79eb91bc46bc75a90bc76dfbeff12355efdc81fa6faec0fc05ace2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-10-19T15:25:43Z","title_canon_sha256":"aa182e2069a16debc2fafd5113005dfd44fd52dc6334e1af7ddefdf283f0f618"},"schema_version":"1.0","source":{"id":"1110.4310","kind":"arxiv","version":1}},"canonical_sha256":"2c5483e1ac21bd37429783acee392e7da4c573e57c2b9853da5994ef433ac42c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2c5483e1ac21bd37429783acee392e7da4c573e57c2b9853da5994ef433ac42c","first_computed_at":"2026-05-18T04:10:37.684531Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:10:37.684531Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"L+nLQko6JxkE3iSo9HG4xkCYcqxygSNIehewtWxdgFXz17oe3tY837l0HsddJh2+U7LYfdFaN36HZrHvQ47iAg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:10:37.685296Z","signed_message":"canonical_sha256_bytes"},"source_id":"1110.4310","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:de39fba6e769a582ccd9bd90c5694d2aefc0fc860751fbb121a46d8f63bab71e","sha256:120893ae837987865f3e512601a4e0f3305a32000eaace2090e24ec1777483e4"],"state_sha256":"1bbb59e4273d0ad098db536bd2c9dbc42a24ed63bfb275e6b86ba9005d043dea"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LtSwJFgpDFEnPyNaiq96ChnWHS7VBl8E6PXUVTszSKz0HEnxoQqpDxC4Bgw3zDhxuCxkotx6e9x0tgaZb8bwCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-07T11:38:41.473606Z","bundle_sha256":"556108879bea37be9d88ad23acd4f2d73a1017fc344850cf441993d9a42abb58"}}