{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:RF4BB32TESMB6VAE6A6XVJVGDG","short_pith_number":"pith:RF4BB32T","canonical_record":{"source":{"id":"1605.04575","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-05-15T16:48:45Z","cross_cats_sorted":[],"title_canon_sha256":"8a03fbd9997646c91f3a736e546f2431ac93a5f24feaacb3fee39df96abe8fe7","abstract_canon_sha256":"e3f2c96a255392997cd49bb944995b7fdd3d8ff60200211a55077595ab54f8e6"},"schema_version":"1.0"},"canonical_sha256":"897810ef5324981f5404f03d7aa6a61986fcee9eaeb2207e21288d3d51105da7","source":{"kind":"arxiv","id":"1605.04575","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1605.04575","created_at":"2026-05-18T01:14:46Z"},{"alias_kind":"arxiv_version","alias_value":"1605.04575v1","created_at":"2026-05-18T01:14:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.04575","created_at":"2026-05-18T01:14:46Z"},{"alias_kind":"pith_short_12","alias_value":"RF4BB32TESMB","created_at":"2026-05-18T12:30:41Z"},{"alias_kind":"pith_short_16","alias_value":"RF4BB32TESMB6VAE","created_at":"2026-05-18T12:30:41Z"},{"alias_kind":"pith_short_8","alias_value":"RF4BB32T","created_at":"2026-05-18T12:30:41Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:RF4BB32TESMB6VAE6A6XVJVGDG","target":"record","payload":{"canonical_record":{"source":{"id":"1605.04575","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-05-15T16:48:45Z","cross_cats_sorted":[],"title_canon_sha256":"8a03fbd9997646c91f3a736e546f2431ac93a5f24feaacb3fee39df96abe8fe7","abstract_canon_sha256":"e3f2c96a255392997cd49bb944995b7fdd3d8ff60200211a55077595ab54f8e6"},"schema_version":"1.0"},"canonical_sha256":"897810ef5324981f5404f03d7aa6a61986fcee9eaeb2207e21288d3d51105da7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:14:46.826141Z","signature_b64":"cGPIZZVbGJ4N/q/eByKFDrtYk59eguEcR1ZSqFUACHCZA32PnrFHzSMj/cD3BgcPse/zOgNRkamwAsVRZgfPDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"897810ef5324981f5404f03d7aa6a61986fcee9eaeb2207e21288d3d51105da7","last_reissued_at":"2026-05-18T01:14:46.825477Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:14:46.825477Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1605.04575","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":"2026-05-18T01:14:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iuBHHbLPqwUUHmopqkVzzOnO6u1VhM3p5/Xy2l4toCkA1rsMOcL40WNekwcLVo0fBV2dPPwlRI9vNml9mOm6CA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T08:08:18.532157Z"},"content_sha256":"ad8c4798612dfbf8c9a7bb14fa8c9baf28affdddd0d0f72fc2d5738a2204463a","schema_version":"1.0","event_id":"sha256:ad8c4798612dfbf8c9a7bb14fa8c9baf28affdddd0d0f72fc2d5738a2204463a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:RF4BB32TESMB6VAE6A6XVJVGDG","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Relating Domination, Exponential Domination, and Porous Exponential Domination","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dieter Rautenbach, Michael A. Henning, Simon J\\\"ager","submitted_at":"2016-05-15T16:48:45Z","abstract_excerpt":"The domination number $\\gamma(G)$ of a graph $G$, its exponential domination number $\\gamma_e(G)$, and its porous exponential domination number $\\gamma_e^*(G)$ satisfy $\\gamma_e^*(G)\\leq \\gamma_e(G)\\leq \\gamma(G)$. We contribute results about the gaps in these inequalities as well as the graphs for which some of the inequalities hold with equality. Relaxing the natural integer linear program whose optimum value is $\\gamma_e^*(G)$, we are led to the definition of the fractional porous exponential domination number $\\gamma_{e,f}^*(G)$ of a graph $G$. For a subcubic tree $T$ of order $n$, we show"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.04575","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":"2026-05-18T01:14:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jCvA8ehfe3l0E8B91iwIiqSZC66ysPmwILxksaBM0yUX35EQVIYLh3e0LOoNoSlUpkZUiCepSZaTF4Zm4Nz/Dg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T08:08:18.532871Z"},"content_sha256":"34d789cc9ed6918f98193379c1f7a0dd1658082bb82d7d55cd22cb67e658d608","schema_version":"1.0","event_id":"sha256:34d789cc9ed6918f98193379c1f7a0dd1658082bb82d7d55cd22cb67e658d608"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RF4BB32TESMB6VAE6A6XVJVGDG/bundle.json","state_url":"https://pith.science/pith/RF4BB32TESMB6VAE6A6XVJVGDG/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RF4BB32TESMB6VAE6A6XVJVGDG/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":"2026-06-11T08:08:18Z","links":{"resolver":"https://pith.science/pith/RF4BB32TESMB6VAE6A6XVJVGDG","bundle":"https://pith.science/pith/RF4BB32TESMB6VAE6A6XVJVGDG/bundle.json","state":"https://pith.science/pith/RF4BB32TESMB6VAE6A6XVJVGDG/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RF4BB32TESMB6VAE6A6XVJVGDG/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:RF4BB32TESMB6VAE6A6XVJVGDG","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":"e3f2c96a255392997cd49bb944995b7fdd3d8ff60200211a55077595ab54f8e6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-05-15T16:48:45Z","title_canon_sha256":"8a03fbd9997646c91f3a736e546f2431ac93a5f24feaacb3fee39df96abe8fe7"},"schema_version":"1.0","source":{"id":"1605.04575","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1605.04575","created_at":"2026-05-18T01:14:46Z"},{"alias_kind":"arxiv_version","alias_value":"1605.04575v1","created_at":"2026-05-18T01:14:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.04575","created_at":"2026-05-18T01:14:46Z"},{"alias_kind":"pith_short_12","alias_value":"RF4BB32TESMB","created_at":"2026-05-18T12:30:41Z"},{"alias_kind":"pith_short_16","alias_value":"RF4BB32TESMB6VAE","created_at":"2026-05-18T12:30:41Z"},{"alias_kind":"pith_short_8","alias_value":"RF4BB32T","created_at":"2026-05-18T12:30:41Z"}],"graph_snapshots":[{"event_id":"sha256:34d789cc9ed6918f98193379c1f7a0dd1658082bb82d7d55cd22cb67e658d608","target":"graph","created_at":"2026-05-18T01:14:46Z","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":"The domination number $\\gamma(G)$ of a graph $G$, its exponential domination number $\\gamma_e(G)$, and its porous exponential domination number $\\gamma_e^*(G)$ satisfy $\\gamma_e^*(G)\\leq \\gamma_e(G)\\leq \\gamma(G)$. We contribute results about the gaps in these inequalities as well as the graphs for which some of the inequalities hold with equality. Relaxing the natural integer linear program whose optimum value is $\\gamma_e^*(G)$, we are led to the definition of the fractional porous exponential domination number $\\gamma_{e,f}^*(G)$ of a graph $G$. For a subcubic tree $T$ of order $n$, we show","authors_text":"Dieter Rautenbach, Michael A. Henning, Simon J\\\"ager","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-05-15T16:48:45Z","title":"Relating Domination, Exponential Domination, and Porous Exponential Domination"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.04575","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:ad8c4798612dfbf8c9a7bb14fa8c9baf28affdddd0d0f72fc2d5738a2204463a","target":"record","created_at":"2026-05-18T01:14:46Z","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":"e3f2c96a255392997cd49bb944995b7fdd3d8ff60200211a55077595ab54f8e6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-05-15T16:48:45Z","title_canon_sha256":"8a03fbd9997646c91f3a736e546f2431ac93a5f24feaacb3fee39df96abe8fe7"},"schema_version":"1.0","source":{"id":"1605.04575","kind":"arxiv","version":1}},"canonical_sha256":"897810ef5324981f5404f03d7aa6a61986fcee9eaeb2207e21288d3d51105da7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"897810ef5324981f5404f03d7aa6a61986fcee9eaeb2207e21288d3d51105da7","first_computed_at":"2026-05-18T01:14:46.825477Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:14:46.825477Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"cGPIZZVbGJ4N/q/eByKFDrtYk59eguEcR1ZSqFUACHCZA32PnrFHzSMj/cD3BgcPse/zOgNRkamwAsVRZgfPDg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:14:46.826141Z","signed_message":"canonical_sha256_bytes"},"source_id":"1605.04575","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ad8c4798612dfbf8c9a7bb14fa8c9baf28affdddd0d0f72fc2d5738a2204463a","sha256:34d789cc9ed6918f98193379c1f7a0dd1658082bb82d7d55cd22cb67e658d608"],"state_sha256":"786fe6482e12834774b2b907dbfb23d81a04857220c055d3d164cb6b6acea13d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"370JIcFoAkyA4eLcKWE45VYyzjqEOuMK72m2m/G7A8vf0uk4GOE84tUAfZjGX6yBAHNbNGJaYjv9PonKIBhyAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-11T08:08:18.537373Z","bundle_sha256":"ff389a1ac2f5c7a8368c8668485d57157b3f9c212ed9716579e4d6197c66c74a"}}