{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:ZGBGNTTKWNBOZU757SQ2RI4BVC","short_pith_number":"pith:ZGBGNTTK","canonical_record":{"source":{"id":"1605.02291","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-05-08T08:15:30Z","cross_cats_sorted":[],"title_canon_sha256":"21f95ccb1822c04a529042e395f296834dc3a3ea954cc0fb7c98ba16392e56dd","abstract_canon_sha256":"ebfa3a20cc349a0fc7f73da19135a0718598ceb2a6518bc63754f1b734bbf7bb"},"schema_version":"1.0"},"canonical_sha256":"c98266ce6ab342ecd3fdfca1a8a381a8adf57cd76823a81a6d7597ae06a81d9b","source":{"kind":"arxiv","id":"1605.02291","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1605.02291","created_at":"2026-05-18T01:15:21Z"},{"alias_kind":"arxiv_version","alias_value":"1605.02291v1","created_at":"2026-05-18T01:15:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.02291","created_at":"2026-05-18T01:15:21Z"},{"alias_kind":"pith_short_12","alias_value":"ZGBGNTTKWNBO","created_at":"2026-05-18T12:30:53Z"},{"alias_kind":"pith_short_16","alias_value":"ZGBGNTTKWNBOZU75","created_at":"2026-05-18T12:30:53Z"},{"alias_kind":"pith_short_8","alias_value":"ZGBGNTTK","created_at":"2026-05-18T12:30:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:ZGBGNTTKWNBOZU757SQ2RI4BVC","target":"record","payload":{"canonical_record":{"source":{"id":"1605.02291","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-05-08T08:15:30Z","cross_cats_sorted":[],"title_canon_sha256":"21f95ccb1822c04a529042e395f296834dc3a3ea954cc0fb7c98ba16392e56dd","abstract_canon_sha256":"ebfa3a20cc349a0fc7f73da19135a0718598ceb2a6518bc63754f1b734bbf7bb"},"schema_version":"1.0"},"canonical_sha256":"c98266ce6ab342ecd3fdfca1a8a381a8adf57cd76823a81a6d7597ae06a81d9b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:15:21.351063Z","signature_b64":"FPZze1tYu7kY5oPjxOca2w9V94FZvR4ZWNpEId7onY6JhHazcPB6rcUuQdnS6sb3xA4Nj3qvc3LRswXGfN0mAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c98266ce6ab342ecd3fdfca1a8a381a8adf57cd76823a81a6d7597ae06a81d9b","last_reissued_at":"2026-05-18T01:15:21.350238Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:15:21.350238Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1605.02291","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":"2026-05-18T01:15:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vpNbmxKooVZ9H/GkBg8sIY7r9Ab8SeU9nbeTFQzJttzuT4fvj5S+CtjYv8FnkxMWIzKevUzV4uNSkxNHLT1aAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T08:57:30.292933Z"},"content_sha256":"5b589ef77cae1c82ced05ad3e61eebbfd8ba3986b0cbba0ce81bdefe8ba0b2b6","schema_version":"1.0","event_id":"sha256:5b589ef77cae1c82ced05ad3e61eebbfd8ba3986b0cbba0ce81bdefe8ba0b2b6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:ZGBGNTTKWNBOZU757SQ2RI4BVC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Domination polynomial of clique cover product of graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Saeid Alikhani, Somayeh Jahari","submitted_at":"2016-05-08T08:15:30Z","abstract_excerpt":"Let $G$ be a simple graph of order $n$. The domination polynomial of $G$ is the polynomial $D(G, x)=\\sum_{i=1}^n d(G,i) x^i$, where $d(G,i)$ is the number of dominating sets of $G$ of size $i$. For two graphs $G$ and $H$, let $\\mathcal{C} = \\{C_1,C_2, \\cdots, C_k\\}$ be a clique cover of $G$ and $U\\subseteq V(H)$. We consider clique cover product which denoted by $G^\\mathcal{C} \\star H^U$ and obtained from $G$ as follows: for each clique $C_i \\in \\mathcal{C}$, add a copy of the graph $H$ and join every vertex of $C_i$ to every vertex of $U$. We prove that the domination polynomial of clique cov"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.02291","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":"2026-05-18T01:15:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GWD1ZAKSOA3xxexgUF5xJEWwz5VWJXSiFtANx4y7vqazcXKfO9LwgQeIYYdO8RXoD34stkFKkYqRKhilyaiJDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T08:57:30.293268Z"},"content_sha256":"db85b63e2150be81c9b44c60863dc4cc3ef40690d0fc38ff84639159cf383a98","schema_version":"1.0","event_id":"sha256:db85b63e2150be81c9b44c60863dc4cc3ef40690d0fc38ff84639159cf383a98"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZGBGNTTKWNBOZU757SQ2RI4BVC/bundle.json","state_url":"https://pith.science/pith/ZGBGNTTKWNBOZU757SQ2RI4BVC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZGBGNTTKWNBOZU757SQ2RI4BVC/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":"2026-06-24T08:57:30Z","links":{"resolver":"https://pith.science/pith/ZGBGNTTKWNBOZU757SQ2RI4BVC","bundle":"https://pith.science/pith/ZGBGNTTKWNBOZU757SQ2RI4BVC/bundle.json","state":"https://pith.science/pith/ZGBGNTTKWNBOZU757SQ2RI4BVC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZGBGNTTKWNBOZU757SQ2RI4BVC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:ZGBGNTTKWNBOZU757SQ2RI4BVC","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":"ebfa3a20cc349a0fc7f73da19135a0718598ceb2a6518bc63754f1b734bbf7bb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-05-08T08:15:30Z","title_canon_sha256":"21f95ccb1822c04a529042e395f296834dc3a3ea954cc0fb7c98ba16392e56dd"},"schema_version":"1.0","source":{"id":"1605.02291","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1605.02291","created_at":"2026-05-18T01:15:21Z"},{"alias_kind":"arxiv_version","alias_value":"1605.02291v1","created_at":"2026-05-18T01:15:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.02291","created_at":"2026-05-18T01:15:21Z"},{"alias_kind":"pith_short_12","alias_value":"ZGBGNTTKWNBO","created_at":"2026-05-18T12:30:53Z"},{"alias_kind":"pith_short_16","alias_value":"ZGBGNTTKWNBOZU75","created_at":"2026-05-18T12:30:53Z"},{"alias_kind":"pith_short_8","alias_value":"ZGBGNTTK","created_at":"2026-05-18T12:30:53Z"}],"graph_snapshots":[{"event_id":"sha256:db85b63e2150be81c9b44c60863dc4cc3ef40690d0fc38ff84639159cf383a98","target":"graph","created_at":"2026-05-18T01:15:21Z","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 simple graph of order $n$. The domination polynomial of $G$ is the polynomial $D(G, x)=\\sum_{i=1}^n d(G,i) x^i$, where $d(G,i)$ is the number of dominating sets of $G$ of size $i$. For two graphs $G$ and $H$, let $\\mathcal{C} = \\{C_1,C_2, \\cdots, C_k\\}$ be a clique cover of $G$ and $U\\subseteq V(H)$. We consider clique cover product which denoted by $G^\\mathcal{C} \\star H^U$ and obtained from $G$ as follows: for each clique $C_i \\in \\mathcal{C}$, add a copy of the graph $H$ and join every vertex of $C_i$ to every vertex of $U$. We prove that the domination polynomial of clique cov","authors_text":"Saeid Alikhani, Somayeh Jahari","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-05-08T08:15:30Z","title":"Domination polynomial of clique cover product of graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.02291","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":"sha256:5b589ef77cae1c82ced05ad3e61eebbfd8ba3986b0cbba0ce81bdefe8ba0b2b6","target":"record","created_at":"2026-05-18T01:15:21Z","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":"ebfa3a20cc349a0fc7f73da19135a0718598ceb2a6518bc63754f1b734bbf7bb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-05-08T08:15:30Z","title_canon_sha256":"21f95ccb1822c04a529042e395f296834dc3a3ea954cc0fb7c98ba16392e56dd"},"schema_version":"1.0","source":{"id":"1605.02291","kind":"arxiv","version":1}},"canonical_sha256":"c98266ce6ab342ecd3fdfca1a8a381a8adf57cd76823a81a6d7597ae06a81d9b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c98266ce6ab342ecd3fdfca1a8a381a8adf57cd76823a81a6d7597ae06a81d9b","first_computed_at":"2026-05-18T01:15:21.350238Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:15:21.350238Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"FPZze1tYu7kY5oPjxOca2w9V94FZvR4ZWNpEId7onY6JhHazcPB6rcUuQdnS6sb3xA4Nj3qvc3LRswXGfN0mAA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:15:21.351063Z","signed_message":"canonical_sha256_bytes"},"source_id":"1605.02291","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5b589ef77cae1c82ced05ad3e61eebbfd8ba3986b0cbba0ce81bdefe8ba0b2b6","sha256:db85b63e2150be81c9b44c60863dc4cc3ef40690d0fc38ff84639159cf383a98"],"state_sha256":"5ee904487d6f2f4145958db7315ae07dbc520419915affb2c07df18ef3cc51de"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vjjRcCwitf68sx2QmARJ7Hs+EWqbu+l6fihi5S4G5Ven24U6APc8hf8TcXyMrTcSSaqOeEkYKHq6dafVmg3kCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T08:57:30.295129Z","bundle_sha256":"0da18ef76f5bb385c992974fe1ed76a7303b34ce30d0784c707ee47782637042"}}