{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:23BJTHYS54425E6ZVY7XRWM4C3","short_pith_number":"pith:23BJTHYS","canonical_record":{"source":{"id":"1112.5047","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-12-21T15:05:55Z","cross_cats_sorted":["math.AC"],"title_canon_sha256":"9a7e378a220e5ea9fcf16d61dbb84d80f66c2c98518454f7db4794e815ad1fe1","abstract_canon_sha256":"a8c27fd2ffedc594301e584e89a5cf1d916328f648d7020a04e590eda5b2ae01"},"schema_version":"1.0"},"canonical_sha256":"d6c2999f12ef39ae93d9ae3f78d99c16d78cf5f17c9abc9e228dd022bed938ca","source":{"kind":"arxiv","id":"1112.5047","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1112.5047","created_at":"2026-05-18T03:49:10Z"},{"alias_kind":"arxiv_version","alias_value":"1112.5047v2","created_at":"2026-05-18T03:49:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1112.5047","created_at":"2026-05-18T03:49:10Z"},{"alias_kind":"pith_short_12","alias_value":"23BJTHYS5442","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_16","alias_value":"23BJTHYS54425E6Z","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_8","alias_value":"23BJTHYS","created_at":"2026-05-18T12:26:18Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:23BJTHYS54425E6ZVY7XRWM4C3","target":"record","payload":{"canonical_record":{"source":{"id":"1112.5047","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-12-21T15:05:55Z","cross_cats_sorted":["math.AC"],"title_canon_sha256":"9a7e378a220e5ea9fcf16d61dbb84d80f66c2c98518454f7db4794e815ad1fe1","abstract_canon_sha256":"a8c27fd2ffedc594301e584e89a5cf1d916328f648d7020a04e590eda5b2ae01"},"schema_version":"1.0"},"canonical_sha256":"d6c2999f12ef39ae93d9ae3f78d99c16d78cf5f17c9abc9e228dd022bed938ca","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:49:10.311872Z","signature_b64":"vSKn1x5NNLj13oYrwIwSyvOUkQhxsDEjU5SlvumL+1oGA4Z1wOVM3lesqPAgSRBAaLBIggfpTSzVvd3Yuo8bAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d6c2999f12ef39ae93d9ae3f78d99c16d78cf5f17c9abc9e228dd022bed938ca","last_reissued_at":"2026-05-18T03:49:10.311442Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:49:10.311442Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1112.5047","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:49:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NZjJS0P6t4Nf3qH55aKAergk5FzZAaRBvDKs3fGU3l8MZC5yM+xil0jl23+YicpCo057C11Q7FmnybKxrSowDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T17:05:35.278339Z"},"content_sha256":"b1198b0268afc762f1e4d42902e4acba5a28adb33dd9b9a406e4ce2e0d24b65b","schema_version":"1.0","event_id":"sha256:b1198b0268afc762f1e4d42902e4acba5a28adb33dd9b9a406e4ce2e0d24b65b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:23BJTHYS54425E6ZVY7XRWM4C3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Separating hyperplanes of edge polytopes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC"],"primary_cat":"math.CO","authors_text":"Nan Li, Takayuki Hibi, Yan X. Zhang","submitted_at":"2011-12-21T15:05:55Z","abstract_excerpt":"Let $G$ be a finite connected simple graph with $d$ vertices and let $\\Pc_G \\subset \\RR^d$ be the edge polytope of $G$. We call $\\Pc_G$ \\emph{decomposable} if $\\Pc_G$ decomposes into integral polytopes $\\Pc_{G^+}$ and $\\Pc_{G^-}$ via a hyperplane. In this paper, we explore various aspects of decomposition of $\\Pc_G$: we give an algorithm deciding the decomposability of $\\Pc_G$, we prove that $\\Pc_G$ is normal if and only if both $\\Pc_{G^+}$ and $\\Pc_{G^-}$ are normal, and we also study how a condition on the toric ideal of $\\Pc_G$ (namely, the ideal being generated by quadratic binomials) beha"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.5047","kind":"arxiv","version":2},"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":"2026-05-18T03:49:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"x/TAHAiU+jVZKoV7RJEE81Bj/SjGbhyx/wDr9GOxHOHlG6IWn2LujC/HfHCMX904qAlW5zS4DsWiyxGbYepjCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T17:05:35.278761Z"},"content_sha256":"a9ecc7f5add6f36780f26afd1573e457c194bbddf488429f42901d2483831c44","schema_version":"1.0","event_id":"sha256:a9ecc7f5add6f36780f26afd1573e457c194bbddf488429f42901d2483831c44"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/23BJTHYS54425E6ZVY7XRWM4C3/bundle.json","state_url":"https://pith.science/pith/23BJTHYS54425E6ZVY7XRWM4C3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/23BJTHYS54425E6ZVY7XRWM4C3/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":"2026-05-28T17:05:35Z","links":{"resolver":"https://pith.science/pith/23BJTHYS54425E6ZVY7XRWM4C3","bundle":"https://pith.science/pith/23BJTHYS54425E6ZVY7XRWM4C3/bundle.json","state":"https://pith.science/pith/23BJTHYS54425E6ZVY7XRWM4C3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/23BJTHYS54425E6ZVY7XRWM4C3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:23BJTHYS54425E6ZVY7XRWM4C3","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":"a8c27fd2ffedc594301e584e89a5cf1d916328f648d7020a04e590eda5b2ae01","cross_cats_sorted":["math.AC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-12-21T15:05:55Z","title_canon_sha256":"9a7e378a220e5ea9fcf16d61dbb84d80f66c2c98518454f7db4794e815ad1fe1"},"schema_version":"1.0","source":{"id":"1112.5047","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1112.5047","created_at":"2026-05-18T03:49:10Z"},{"alias_kind":"arxiv_version","alias_value":"1112.5047v2","created_at":"2026-05-18T03:49:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1112.5047","created_at":"2026-05-18T03:49:10Z"},{"alias_kind":"pith_short_12","alias_value":"23BJTHYS5442","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_16","alias_value":"23BJTHYS54425E6Z","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_8","alias_value":"23BJTHYS","created_at":"2026-05-18T12:26:18Z"}],"graph_snapshots":[{"event_id":"sha256:a9ecc7f5add6f36780f26afd1573e457c194bbddf488429f42901d2483831c44","target":"graph","created_at":"2026-05-18T03:49:10Z","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 finite connected simple graph with $d$ vertices and let $\\Pc_G \\subset \\RR^d$ be the edge polytope of $G$. We call $\\Pc_G$ \\emph{decomposable} if $\\Pc_G$ decomposes into integral polytopes $\\Pc_{G^+}$ and $\\Pc_{G^-}$ via a hyperplane. In this paper, we explore various aspects of decomposition of $\\Pc_G$: we give an algorithm deciding the decomposability of $\\Pc_G$, we prove that $\\Pc_G$ is normal if and only if both $\\Pc_{G^+}$ and $\\Pc_{G^-}$ are normal, and we also study how a condition on the toric ideal of $\\Pc_G$ (namely, the ideal being generated by quadratic binomials) beha","authors_text":"Nan Li, Takayuki Hibi, Yan X. Zhang","cross_cats":["math.AC"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-12-21T15:05:55Z","title":"Separating hyperplanes of edge polytopes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.5047","kind":"arxiv","version":2},"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:b1198b0268afc762f1e4d42902e4acba5a28adb33dd9b9a406e4ce2e0d24b65b","target":"record","created_at":"2026-05-18T03:49:10Z","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":"a8c27fd2ffedc594301e584e89a5cf1d916328f648d7020a04e590eda5b2ae01","cross_cats_sorted":["math.AC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-12-21T15:05:55Z","title_canon_sha256":"9a7e378a220e5ea9fcf16d61dbb84d80f66c2c98518454f7db4794e815ad1fe1"},"schema_version":"1.0","source":{"id":"1112.5047","kind":"arxiv","version":2}},"canonical_sha256":"d6c2999f12ef39ae93d9ae3f78d99c16d78cf5f17c9abc9e228dd022bed938ca","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d6c2999f12ef39ae93d9ae3f78d99c16d78cf5f17c9abc9e228dd022bed938ca","first_computed_at":"2026-05-18T03:49:10.311442Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:49:10.311442Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"vSKn1x5NNLj13oYrwIwSyvOUkQhxsDEjU5SlvumL+1oGA4Z1wOVM3lesqPAgSRBAaLBIggfpTSzVvd3Yuo8bAA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:49:10.311872Z","signed_message":"canonical_sha256_bytes"},"source_id":"1112.5047","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b1198b0268afc762f1e4d42902e4acba5a28adb33dd9b9a406e4ce2e0d24b65b","sha256:a9ecc7f5add6f36780f26afd1573e457c194bbddf488429f42901d2483831c44"],"state_sha256":"d728a02e995a2deeadb51ce07dd4508bb3211a010265668a6f74d73195660ce1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4brLUC+bZx4wZZheaUi9Ap0ocSF67ADtC7Yp9+4c7JotGJlpyD2eRVANyqhwH0/OjhwgUMZLFiaiCnWbHWmiCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T17:05:35.281023Z","bundle_sha256":"0859d478d5648187a2921c0ccccf8c6c0f60dc2c54d0ba6560bf9ec6937cb289"}}