{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:P7YZ4EIW5CXRLK3MPVIEAKHSHA","short_pith_number":"pith:P7YZ4EIW","canonical_record":{"source":{"id":"1906.00073","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-05-31T20:45:25Z","cross_cats_sorted":[],"title_canon_sha256":"a4cfb50401f216e97b0f98841adba4cd031fdfbb6b6725aeff32af2b7970182f","abstract_canon_sha256":"e4248447e0113a9c57710f12c0b78f7778434384ee1dbdb08155a45ee55cf4c8"},"schema_version":"1.0"},"canonical_sha256":"7ff19e1116e8af15ab6c7d504028f2382347ad392f58e2ef92414dd6cd1af5d4","source":{"kind":"arxiv","id":"1906.00073","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1906.00073","created_at":"2026-05-17T23:44:33Z"},{"alias_kind":"arxiv_version","alias_value":"1906.00073v1","created_at":"2026-05-17T23:44:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1906.00073","created_at":"2026-05-17T23:44:33Z"},{"alias_kind":"pith_short_12","alias_value":"P7YZ4EIW5CXR","created_at":"2026-05-18T12:33:24Z"},{"alias_kind":"pith_short_16","alias_value":"P7YZ4EIW5CXRLK3M","created_at":"2026-05-18T12:33:24Z"},{"alias_kind":"pith_short_8","alias_value":"P7YZ4EIW","created_at":"2026-05-18T12:33:24Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:P7YZ4EIW5CXRLK3MPVIEAKHSHA","target":"record","payload":{"canonical_record":{"source":{"id":"1906.00073","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-05-31T20:45:25Z","cross_cats_sorted":[],"title_canon_sha256":"a4cfb50401f216e97b0f98841adba4cd031fdfbb6b6725aeff32af2b7970182f","abstract_canon_sha256":"e4248447e0113a9c57710f12c0b78f7778434384ee1dbdb08155a45ee55cf4c8"},"schema_version":"1.0"},"canonical_sha256":"7ff19e1116e8af15ab6c7d504028f2382347ad392f58e2ef92414dd6cd1af5d4","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:44:33.537585Z","signature_b64":"b+GirA1WUvs+Up2vAIallY1EST5C5lm7uxgXfCieo2isrImQHOsE/0s4GdrVpsl/b/xSD3XEcQSyqSU2QYp1BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7ff19e1116e8af15ab6c7d504028f2382347ad392f58e2ef92414dd6cd1af5d4","last_reissued_at":"2026-05-17T23:44:33.536999Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:44:33.536999Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1906.00073","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":"2026-05-17T23:44:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vhadkw69b8x/TOJgZeev9pfxg0Dw5m5CFIAZVbsGoKCbWXlq2GdriiI57U/d2AmaATRbuB3vk6lGr8ETHxyMDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T22:21:59.320005Z"},"content_sha256":"a56ea904737b9d2493f4980ba3a5251ada53be4313f635cb05347a34d0ac84da","schema_version":"1.0","event_id":"sha256:a56ea904737b9d2493f4980ba3a5251ada53be4313f635cb05347a34d0ac84da"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:P7YZ4EIW5CXRLK3MPVIEAKHSHA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"$\\beta$-Packing Sets in Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Benjamin M. Case, Evan M. Haithcock, Renu C. Laskar","submitted_at":"2019-05-31T20:45:25Z","abstract_excerpt":"A set $S\\subseteq V$ is $\\alpha$-dominating if for all $v\\in V-S$, $|N(v) \\cap S | \\geq \\alpha |N(v)|.$ The $\\alpha$-domination number of $G$ equals the minimum cardinality of an $\\alpha$-dominating set $S$ in $G$. Since being introduced by Dunbar, et al. in 2000, $\\alpha$-domination has been studied for various graphs and a variety of bounds have been developed. In this paper, we propose a new parameter derived by flipping the inequality in the definition of $\\alpha$-domination. We say a set $S \\subset V$ is a $\\beta$-packing set of a graph $G$ if $S$ is a proper, maximal set having the prope"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.00073","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":"2026-05-17T23:44:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PpEMfGoaXEPEuLL+KdpL+xC1cZe4b9y88VY8EFDj8bialPz2P7c1Oy+8hJ4v8jEfpMMHc+P8swk+iELyy+iFBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T22:21:59.320789Z"},"content_sha256":"7126f6ba40ca7d48d3fc64f58d52a5c87c518f30e18c4b4e7afdace48b8301f5","schema_version":"1.0","event_id":"sha256:7126f6ba40ca7d48d3fc64f58d52a5c87c518f30e18c4b4e7afdace48b8301f5"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/P7YZ4EIW5CXRLK3MPVIEAKHSHA/bundle.json","state_url":"https://pith.science/pith/P7YZ4EIW5CXRLK3MPVIEAKHSHA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/P7YZ4EIW5CXRLK3MPVIEAKHSHA/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":"2026-06-11T22:21:59Z","links":{"resolver":"https://pith.science/pith/P7YZ4EIW5CXRLK3MPVIEAKHSHA","bundle":"https://pith.science/pith/P7YZ4EIW5CXRLK3MPVIEAKHSHA/bundle.json","state":"https://pith.science/pith/P7YZ4EIW5CXRLK3MPVIEAKHSHA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/P7YZ4EIW5CXRLK3MPVIEAKHSHA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:P7YZ4EIW5CXRLK3MPVIEAKHSHA","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":"e4248447e0113a9c57710f12c0b78f7778434384ee1dbdb08155a45ee55cf4c8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-05-31T20:45:25Z","title_canon_sha256":"a4cfb50401f216e97b0f98841adba4cd031fdfbb6b6725aeff32af2b7970182f"},"schema_version":"1.0","source":{"id":"1906.00073","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1906.00073","created_at":"2026-05-17T23:44:33Z"},{"alias_kind":"arxiv_version","alias_value":"1906.00073v1","created_at":"2026-05-17T23:44:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1906.00073","created_at":"2026-05-17T23:44:33Z"},{"alias_kind":"pith_short_12","alias_value":"P7YZ4EIW5CXR","created_at":"2026-05-18T12:33:24Z"},{"alias_kind":"pith_short_16","alias_value":"P7YZ4EIW5CXRLK3M","created_at":"2026-05-18T12:33:24Z"},{"alias_kind":"pith_short_8","alias_value":"P7YZ4EIW","created_at":"2026-05-18T12:33:24Z"}],"graph_snapshots":[{"event_id":"sha256:7126f6ba40ca7d48d3fc64f58d52a5c87c518f30e18c4b4e7afdace48b8301f5","target":"graph","created_at":"2026-05-17T23:44:33Z","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":"A set $S\\subseteq V$ is $\\alpha$-dominating if for all $v\\in V-S$, $|N(v) \\cap S | \\geq \\alpha |N(v)|.$ The $\\alpha$-domination number of $G$ equals the minimum cardinality of an $\\alpha$-dominating set $S$ in $G$. Since being introduced by Dunbar, et al. in 2000, $\\alpha$-domination has been studied for various graphs and a variety of bounds have been developed. In this paper, we propose a new parameter derived by flipping the inequality in the definition of $\\alpha$-domination. We say a set $S \\subset V$ is a $\\beta$-packing set of a graph $G$ if $S$ is a proper, maximal set having the prope","authors_text":"Benjamin M. Case, Evan M. Haithcock, Renu C. Laskar","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-05-31T20:45:25Z","title":"$\\beta$-Packing Sets in Graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.00073","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:a56ea904737b9d2493f4980ba3a5251ada53be4313f635cb05347a34d0ac84da","target":"record","created_at":"2026-05-17T23:44:33Z","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":"e4248447e0113a9c57710f12c0b78f7778434384ee1dbdb08155a45ee55cf4c8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-05-31T20:45:25Z","title_canon_sha256":"a4cfb50401f216e97b0f98841adba4cd031fdfbb6b6725aeff32af2b7970182f"},"schema_version":"1.0","source":{"id":"1906.00073","kind":"arxiv","version":1}},"canonical_sha256":"7ff19e1116e8af15ab6c7d504028f2382347ad392f58e2ef92414dd6cd1af5d4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7ff19e1116e8af15ab6c7d504028f2382347ad392f58e2ef92414dd6cd1af5d4","first_computed_at":"2026-05-17T23:44:33.536999Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:44:33.536999Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"b+GirA1WUvs+Up2vAIallY1EST5C5lm7uxgXfCieo2isrImQHOsE/0s4GdrVpsl/b/xSD3XEcQSyqSU2QYp1BA==","signature_status":"signed_v1","signed_at":"2026-05-17T23:44:33.537585Z","signed_message":"canonical_sha256_bytes"},"source_id":"1906.00073","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a56ea904737b9d2493f4980ba3a5251ada53be4313f635cb05347a34d0ac84da","sha256:7126f6ba40ca7d48d3fc64f58d52a5c87c518f30e18c4b4e7afdace48b8301f5"],"state_sha256":"67cfb9886afea63995ed0bf6aefc7467bd535ba887dc76fe7684bb358826b9d4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RUo76QSjLxdlsRfsENmctTYXZ4rKTXbpIqgfKKR0DciTfeVetN6+YB7GJnn7mFPa1rWNjkdRVptWAWGWTqomBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-11T22:21:59.325303Z","bundle_sha256":"50ea94470f5f7d6414f557fd8cca668c2acd8a372d5c2722e239df711124da25"}}