{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:GRJIJBDBYXCTOBDZ7FJYG24YPB","short_pith_number":"pith:GRJIJBDB","canonical_record":{"source":{"id":"1701.03453","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-01-12T18:54:42Z","cross_cats_sorted":[],"title_canon_sha256":"6601b4430ddeca1dfafb2ca2eea480d19b07e2fddce6242e56f440213835d754","abstract_canon_sha256":"8c941fccb7874f7efa249245f7dc50f0ea7387264973eda94b0aad6ffbdf67f7"},"schema_version":"1.0"},"canonical_sha256":"3452848461c5c5370479f953836b987874e0c89fb4ae99e2130e032fdb811188","source":{"kind":"arxiv","id":"1701.03453","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1701.03453","created_at":"2026-05-18T00:52:55Z"},{"alias_kind":"arxiv_version","alias_value":"1701.03453v1","created_at":"2026-05-18T00:52:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1701.03453","created_at":"2026-05-18T00:52:55Z"},{"alias_kind":"pith_short_12","alias_value":"GRJIJBDBYXCT","created_at":"2026-05-18T12:31:18Z"},{"alias_kind":"pith_short_16","alias_value":"GRJIJBDBYXCTOBDZ","created_at":"2026-05-18T12:31:18Z"},{"alias_kind":"pith_short_8","alias_value":"GRJIJBDB","created_at":"2026-05-18T12:31:18Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:GRJIJBDBYXCTOBDZ7FJYG24YPB","target":"record","payload":{"canonical_record":{"source":{"id":"1701.03453","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-01-12T18:54:42Z","cross_cats_sorted":[],"title_canon_sha256":"6601b4430ddeca1dfafb2ca2eea480d19b07e2fddce6242e56f440213835d754","abstract_canon_sha256":"8c941fccb7874f7efa249245f7dc50f0ea7387264973eda94b0aad6ffbdf67f7"},"schema_version":"1.0"},"canonical_sha256":"3452848461c5c5370479f953836b987874e0c89fb4ae99e2130e032fdb811188","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:52:55.810803Z","signature_b64":"8PBV5OJm4qOPf5voSW0f0ZfHrtwtjHO908sxL25Ov0w0ms5uu9Stqa9Ah4utuHWergF3FnGZdcqc8dUlytRtBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3452848461c5c5370479f953836b987874e0c89fb4ae99e2130e032fdb811188","last_reissued_at":"2026-05-18T00:52:55.810392Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:52:55.810392Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1701.03453","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":"2026-05-18T00:52:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YFsvxYS/CdHDy/bLYD+9PoAAmIvcEEKs1Z19PD7v7QlmExuTiHTPGs0QTvcBQ2pGcfhMUSuGn1OgbrkYpdCPBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T00:22:48.741370Z"},"content_sha256":"910c83805bbe64b949a414172688d5779cbe5557701000b66e57fe8816dd5c7c","schema_version":"1.0","event_id":"sha256:910c83805bbe64b949a414172688d5779cbe5557701000b66e57fe8816dd5c7c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:GRJIJBDBYXCTOBDZ7FJYG24YPB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Counting Dominating Sets of Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Irene Heinrich, Peter Tittmann","submitted_at":"2017-01-12T18:54:42Z","abstract_excerpt":"Counting dominating sets in a graph $G$ is closely related to the neighborhood complex of $G$. We exploit this relation to prove that the number of dominating sets $d(G)$ of a graph is determined by the number of complete bipartite subgraphs of its complement. More precisely, we state the following. Let $G$ be a simple graph of order $n$ such that its complement has exactly $a(G)$ subgraphs isomorphic to $K_{2p,2q}$ and exactly $b(G)$ subgraphs isomorphic to $K_{2p+1,2q+1}$. Then $d(G) = 2^n -1 + 2[a(G)-b(G)]$. We also show some new relations between the domination polynomial and the neighborh"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.03453","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":"2026-05-18T00:52:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"11Qmmjxksw5aIb6h8ekxfzq9fudc9tnQI2bGQn/cY4bl/esm0meJiPDknwbXT4hc8rNtebFjaqP+ElKMbzhGDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T00:22:48.742072Z"},"content_sha256":"59c206456cb0b262b502141d3930737adf608a9e44cfc7ff17e8226b82fdadd9","schema_version":"1.0","event_id":"sha256:59c206456cb0b262b502141d3930737adf608a9e44cfc7ff17e8226b82fdadd9"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GRJIJBDBYXCTOBDZ7FJYG24YPB/bundle.json","state_url":"https://pith.science/pith/GRJIJBDBYXCTOBDZ7FJYG24YPB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GRJIJBDBYXCTOBDZ7FJYG24YPB/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":"2026-06-08T00:22:48Z","links":{"resolver":"https://pith.science/pith/GRJIJBDBYXCTOBDZ7FJYG24YPB","bundle":"https://pith.science/pith/GRJIJBDBYXCTOBDZ7FJYG24YPB/bundle.json","state":"https://pith.science/pith/GRJIJBDBYXCTOBDZ7FJYG24YPB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GRJIJBDBYXCTOBDZ7FJYG24YPB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:GRJIJBDBYXCTOBDZ7FJYG24YPB","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":"8c941fccb7874f7efa249245f7dc50f0ea7387264973eda94b0aad6ffbdf67f7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-01-12T18:54:42Z","title_canon_sha256":"6601b4430ddeca1dfafb2ca2eea480d19b07e2fddce6242e56f440213835d754"},"schema_version":"1.0","source":{"id":"1701.03453","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1701.03453","created_at":"2026-05-18T00:52:55Z"},{"alias_kind":"arxiv_version","alias_value":"1701.03453v1","created_at":"2026-05-18T00:52:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1701.03453","created_at":"2026-05-18T00:52:55Z"},{"alias_kind":"pith_short_12","alias_value":"GRJIJBDBYXCT","created_at":"2026-05-18T12:31:18Z"},{"alias_kind":"pith_short_16","alias_value":"GRJIJBDBYXCTOBDZ","created_at":"2026-05-18T12:31:18Z"},{"alias_kind":"pith_short_8","alias_value":"GRJIJBDB","created_at":"2026-05-18T12:31:18Z"}],"graph_snapshots":[{"event_id":"sha256:59c206456cb0b262b502141d3930737adf608a9e44cfc7ff17e8226b82fdadd9","target":"graph","created_at":"2026-05-18T00:52:55Z","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":"Counting dominating sets in a graph $G$ is closely related to the neighborhood complex of $G$. We exploit this relation to prove that the number of dominating sets $d(G)$ of a graph is determined by the number of complete bipartite subgraphs of its complement. More precisely, we state the following. Let $G$ be a simple graph of order $n$ such that its complement has exactly $a(G)$ subgraphs isomorphic to $K_{2p,2q}$ and exactly $b(G)$ subgraphs isomorphic to $K_{2p+1,2q+1}$. Then $d(G) = 2^n -1 + 2[a(G)-b(G)]$. We also show some new relations between the domination polynomial and the neighborh","authors_text":"Irene Heinrich, Peter Tittmann","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-01-12T18:54:42Z","title":"Counting Dominating Sets of Graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.03453","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":"sha256:910c83805bbe64b949a414172688d5779cbe5557701000b66e57fe8816dd5c7c","target":"record","created_at":"2026-05-18T00:52:55Z","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":"8c941fccb7874f7efa249245f7dc50f0ea7387264973eda94b0aad6ffbdf67f7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-01-12T18:54:42Z","title_canon_sha256":"6601b4430ddeca1dfafb2ca2eea480d19b07e2fddce6242e56f440213835d754"},"schema_version":"1.0","source":{"id":"1701.03453","kind":"arxiv","version":1}},"canonical_sha256":"3452848461c5c5370479f953836b987874e0c89fb4ae99e2130e032fdb811188","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3452848461c5c5370479f953836b987874e0c89fb4ae99e2130e032fdb811188","first_computed_at":"2026-05-18T00:52:55.810392Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:52:55.810392Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8PBV5OJm4qOPf5voSW0f0ZfHrtwtjHO908sxL25Ov0w0ms5uu9Stqa9Ah4utuHWergF3FnGZdcqc8dUlytRtBw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:52:55.810803Z","signed_message":"canonical_sha256_bytes"},"source_id":"1701.03453","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:910c83805bbe64b949a414172688d5779cbe5557701000b66e57fe8816dd5c7c","sha256:59c206456cb0b262b502141d3930737adf608a9e44cfc7ff17e8226b82fdadd9"],"state_sha256":"68c2da5622dba3497c3e790a203abddacb9e5c709b0eb9b5d2dc3d023cda7532"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eJpwSVXdy1kTszEVDzTYA9uXAISCWGsG/ET6TqoEikb2jbCfiyJxhKZSflc9X3jEi3/3IrusiP1zmVX5qAsABg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T00:22:48.745801Z","bundle_sha256":"51d1d431c1769c1a9bf0dacca0703d2b3e0aa48ccb2ac7e2c938a809997a7e6d"}}