{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:3NGORA2UXSMMGDUBPQP2O7AHQP","short_pith_number":"pith:3NGORA2U","canonical_record":{"source":{"id":"1512.01067","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-12-03T13:07:03Z","cross_cats_sorted":[],"title_canon_sha256":"d3023506afa218077d8bf9aa42034e5a6580f90e09b841ab9b77c76f1c166a2b","abstract_canon_sha256":"06c372a768f3a036d737dad5f824c135b4d67ae675056a49eb865ef5b32fdc43"},"schema_version":"1.0"},"canonical_sha256":"db4ce88354bc98c30e817c1fa77c0783fecc7b9a5436802bca0368403399d402","source":{"kind":"arxiv","id":"1512.01067","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1512.01067","created_at":"2026-05-18T01:25:19Z"},{"alias_kind":"arxiv_version","alias_value":"1512.01067v1","created_at":"2026-05-18T01:25:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.01067","created_at":"2026-05-18T01:25:19Z"},{"alias_kind":"pith_short_12","alias_value":"3NGORA2UXSMM","created_at":"2026-05-18T12:29:02Z"},{"alias_kind":"pith_short_16","alias_value":"3NGORA2UXSMMGDUB","created_at":"2026-05-18T12:29:02Z"},{"alias_kind":"pith_short_8","alias_value":"3NGORA2U","created_at":"2026-05-18T12:29:02Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:3NGORA2UXSMMGDUBPQP2O7AHQP","target":"record","payload":{"canonical_record":{"source":{"id":"1512.01067","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-12-03T13:07:03Z","cross_cats_sorted":[],"title_canon_sha256":"d3023506afa218077d8bf9aa42034e5a6580f90e09b841ab9b77c76f1c166a2b","abstract_canon_sha256":"06c372a768f3a036d737dad5f824c135b4d67ae675056a49eb865ef5b32fdc43"},"schema_version":"1.0"},"canonical_sha256":"db4ce88354bc98c30e817c1fa77c0783fecc7b9a5436802bca0368403399d402","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:25:19.018109Z","signature_b64":"uO7fGDMuyvExBHJsNY7mg3m0DiWSuqAeFkdd7OvviLGxgulWA3NrojQVm8YyTh9zp+C7JnBYwfqQiPVmrQhjBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"db4ce88354bc98c30e817c1fa77c0783fecc7b9a5436802bca0368403399d402","last_reissued_at":"2026-05-18T01:25:19.017481Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:25:19.017481Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1512.01067","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":"2026-05-18T01:25:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0wHFuJpb+bs4bV0ixlGBgfjKQCSmLifsBlRDhII+8mNS0utLn+BdI6Cor3bTJmRT86e4Jj3TpiYFQzjNHRSBCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T03:11:38.525745Z"},"content_sha256":"35704842c434c2c97684ef233444653e1778593c14a83271057f967d19f7e451","schema_version":"1.0","event_id":"sha256:35704842c434c2c97684ef233444653e1778593c14a83271057f967d19f7e451"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:3NGORA2UXSMMGDUBPQP2O7AHQP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Relating $2$-Rainbow Domination to Roman domination","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dieter Rautenbach, Jos\\'e D. Alvarado, Simone Dantas","submitted_at":"2015-12-03T13:07:03Z","abstract_excerpt":"For a graph $G$, let $\\gamma_R(G)$ and $\\gamma_{r2}(G)$ denote the Roman domination number of $G$ and the $2$-rainbow domination number of $G$, respectively. It is known that $\\gamma_{r2}(G)\\leq \\gamma_R(G)\\leq \\frac{3}{2}\\gamma_{r2}(G)$. Fujita and Furuya (Difference between 2-rainbow domination and Roman domination in graphs, Discrete Applied Mathematics 161 (2013) 806-812) present some kind of characterization of the graphs $G$ for which $\\gamma_R(G)-\\gamma_{r2}(G)=k$ for some integer $k$. Unfortunately, their result does not lead to an algorithm that allows to recognize these graphs effici"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.01067","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":"2026-05-18T01:25:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"H/PgBRzXfjesJ6j3kBXrzdtCER0TLeoHR7tz5hEb4Ols4I4tFI3E8tNmT/S974+0QG+yTYwKFg0BBAf6CXGQDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T03:11:38.526086Z"},"content_sha256":"dc2b5846d3902f3a0f652c63e7cd73bedb76bbc6b02c671e261e4bd850e470b5","schema_version":"1.0","event_id":"sha256:dc2b5846d3902f3a0f652c63e7cd73bedb76bbc6b02c671e261e4bd850e470b5"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3NGORA2UXSMMGDUBPQP2O7AHQP/bundle.json","state_url":"https://pith.science/pith/3NGORA2UXSMMGDUBPQP2O7AHQP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3NGORA2UXSMMGDUBPQP2O7AHQP/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":"2026-06-24T03:11:38Z","links":{"resolver":"https://pith.science/pith/3NGORA2UXSMMGDUBPQP2O7AHQP","bundle":"https://pith.science/pith/3NGORA2UXSMMGDUBPQP2O7AHQP/bundle.json","state":"https://pith.science/pith/3NGORA2UXSMMGDUBPQP2O7AHQP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3NGORA2UXSMMGDUBPQP2O7AHQP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:3NGORA2UXSMMGDUBPQP2O7AHQP","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":"06c372a768f3a036d737dad5f824c135b4d67ae675056a49eb865ef5b32fdc43","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-12-03T13:07:03Z","title_canon_sha256":"d3023506afa218077d8bf9aa42034e5a6580f90e09b841ab9b77c76f1c166a2b"},"schema_version":"1.0","source":{"id":"1512.01067","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1512.01067","created_at":"2026-05-18T01:25:19Z"},{"alias_kind":"arxiv_version","alias_value":"1512.01067v1","created_at":"2026-05-18T01:25:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.01067","created_at":"2026-05-18T01:25:19Z"},{"alias_kind":"pith_short_12","alias_value":"3NGORA2UXSMM","created_at":"2026-05-18T12:29:02Z"},{"alias_kind":"pith_short_16","alias_value":"3NGORA2UXSMMGDUB","created_at":"2026-05-18T12:29:02Z"},{"alias_kind":"pith_short_8","alias_value":"3NGORA2U","created_at":"2026-05-18T12:29:02Z"}],"graph_snapshots":[{"event_id":"sha256:dc2b5846d3902f3a0f652c63e7cd73bedb76bbc6b02c671e261e4bd850e470b5","target":"graph","created_at":"2026-05-18T01:25:19Z","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 graph $G$, let $\\gamma_R(G)$ and $\\gamma_{r2}(G)$ denote the Roman domination number of $G$ and the $2$-rainbow domination number of $G$, respectively. It is known that $\\gamma_{r2}(G)\\leq \\gamma_R(G)\\leq \\frac{3}{2}\\gamma_{r2}(G)$. Fujita and Furuya (Difference between 2-rainbow domination and Roman domination in graphs, Discrete Applied Mathematics 161 (2013) 806-812) present some kind of characterization of the graphs $G$ for which $\\gamma_R(G)-\\gamma_{r2}(G)=k$ for some integer $k$. Unfortunately, their result does not lead to an algorithm that allows to recognize these graphs effici","authors_text":"Dieter Rautenbach, Jos\\'e D. Alvarado, Simone Dantas","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-12-03T13:07:03Z","title":"Relating $2$-Rainbow Domination to Roman domination"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.01067","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":"sha256:35704842c434c2c97684ef233444653e1778593c14a83271057f967d19f7e451","target":"record","created_at":"2026-05-18T01:25:19Z","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":"06c372a768f3a036d737dad5f824c135b4d67ae675056a49eb865ef5b32fdc43","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-12-03T13:07:03Z","title_canon_sha256":"d3023506afa218077d8bf9aa42034e5a6580f90e09b841ab9b77c76f1c166a2b"},"schema_version":"1.0","source":{"id":"1512.01067","kind":"arxiv","version":1}},"canonical_sha256":"db4ce88354bc98c30e817c1fa77c0783fecc7b9a5436802bca0368403399d402","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"db4ce88354bc98c30e817c1fa77c0783fecc7b9a5436802bca0368403399d402","first_computed_at":"2026-05-18T01:25:19.017481Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:25:19.017481Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"uO7fGDMuyvExBHJsNY7mg3m0DiWSuqAeFkdd7OvviLGxgulWA3NrojQVm8YyTh9zp+C7JnBYwfqQiPVmrQhjBg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:25:19.018109Z","signed_message":"canonical_sha256_bytes"},"source_id":"1512.01067","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:35704842c434c2c97684ef233444653e1778593c14a83271057f967d19f7e451","sha256:dc2b5846d3902f3a0f652c63e7cd73bedb76bbc6b02c671e261e4bd850e470b5"],"state_sha256":"165865b5591bfd05ca6c462b3c66552c35b4cdebdc1fa42398f8f919f6b08d20"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FtrY35f8Gmq555QBH+OicfQDc+bvsn840g8S5++PPjQqWom/xdKf5mMF/o8aEeLZcNEVim3RE7dOlAIqQVkoCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T03:11:38.528001Z","bundle_sha256":"a06e260a25426f29e6a6c010eb3028d5bde84e69f8547c25f5826d07db5b4e33"}}