{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:XPQRMLEBCGINZ7GUZQZWJAG2K5","short_pith_number":"pith:XPQRMLEB","canonical_record":{"source":{"id":"1206.5926","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-06-26T08:58:50Z","cross_cats_sorted":[],"title_canon_sha256":"e7dda51325b02b529fe2d517d7d20b511fa2f95ba08b7275753990941f84bf54","abstract_canon_sha256":"84f71f3a687988071fd1c8fe513558be6f44a84e0ac080b68352c04c86705171"},"schema_version":"1.0"},"canonical_sha256":"bbe1162c811190dcfcd4cc336480da5774a0f3eb61f0201ae4a8e2e0b99b8150","source":{"kind":"arxiv","id":"1206.5926","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1206.5926","created_at":"2026-05-18T03:27:11Z"},{"alias_kind":"arxiv_version","alias_value":"1206.5926v2","created_at":"2026-05-18T03:27:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.5926","created_at":"2026-05-18T03:27:11Z"},{"alias_kind":"pith_short_12","alias_value":"XPQRMLEBCGIN","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_16","alias_value":"XPQRMLEBCGINZ7GU","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_8","alias_value":"XPQRMLEB","created_at":"2026-05-18T12:27:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:XPQRMLEBCGINZ7GUZQZWJAG2K5","target":"record","payload":{"canonical_record":{"source":{"id":"1206.5926","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-06-26T08:58:50Z","cross_cats_sorted":[],"title_canon_sha256":"e7dda51325b02b529fe2d517d7d20b511fa2f95ba08b7275753990941f84bf54","abstract_canon_sha256":"84f71f3a687988071fd1c8fe513558be6f44a84e0ac080b68352c04c86705171"},"schema_version":"1.0"},"canonical_sha256":"bbe1162c811190dcfcd4cc336480da5774a0f3eb61f0201ae4a8e2e0b99b8150","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:27:11.443199Z","signature_b64":"l489/IWE8M6U7bfZ9uAonVUXuCIr1EWRQMQQqseB4OmKZkSQBAmj7jCXNtEcOMNMN08EXnjN/hIyIZSx2ivXBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bbe1162c811190dcfcd4cc336480da5774a0f3eb61f0201ae4a8e2e0b99b8150","last_reissued_at":"2026-05-18T03:27:11.442616Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:27:11.442616Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1206.5926","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":"2026-05-18T03:27:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mEvHaPCkQzBfG5PP9kiqJT9MaRfe0o39HvBw3secgOEmE6LXGt9Af0AiwV60eCQB1kXFu7833zDxCxwaNh5BAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T01:41:53.143506Z"},"content_sha256":"1492659303c1f4fc8e499b0e3fea4d78f702863a4fdd83512e9e73383e241718","schema_version":"1.0","event_id":"sha256:1492659303c1f4fc8e499b0e3fea4d78f702863a4fdd83512e9e73383e241718"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:XPQRMLEBCGINZ7GUZQZWJAG2K5","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Recurrence relations and splitting formulas for the domination polynomial","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Frank Simon, James Preen, Martin Trinks, Peter Tittmann, Tomer Kotek","submitted_at":"2012-06-26T08:58:50Z","abstract_excerpt":"The domination polynomial D(G,x) of a graph G is the generating function of its dominating sets. We prove that D(G,x) satisfies a wide range of reduction formulas. We show linear recurrence relations for D(G,x) for arbitrary graphs and for various special cases. We give splitting formulas for D(G,x) based on articulation vertices, and more generally, on splitting sets of vertices."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.5926","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":"2026-05-18T03:27:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"L3R3LpsFrUF48Ot3l4tuUdmdEfIOxSuZ6ebBHWb0zf83DnYW5UUEHHjK8gqVODjeTyPROiP2BXWMa7Q0gurBAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T01:41:53.144144Z"},"content_sha256":"cc9b30f5a4bd193e28eb14dbd1f308b9c756572b379b38157e4ced41156c0baf","schema_version":"1.0","event_id":"sha256:cc9b30f5a4bd193e28eb14dbd1f308b9c756572b379b38157e4ced41156c0baf"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XPQRMLEBCGINZ7GUZQZWJAG2K5/bundle.json","state_url":"https://pith.science/pith/XPQRMLEBCGINZ7GUZQZWJAG2K5/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XPQRMLEBCGINZ7GUZQZWJAG2K5/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":"2026-06-08T01:41:53Z","links":{"resolver":"https://pith.science/pith/XPQRMLEBCGINZ7GUZQZWJAG2K5","bundle":"https://pith.science/pith/XPQRMLEBCGINZ7GUZQZWJAG2K5/bundle.json","state":"https://pith.science/pith/XPQRMLEBCGINZ7GUZQZWJAG2K5/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XPQRMLEBCGINZ7GUZQZWJAG2K5/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:XPQRMLEBCGINZ7GUZQZWJAG2K5","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":"84f71f3a687988071fd1c8fe513558be6f44a84e0ac080b68352c04c86705171","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-06-26T08:58:50Z","title_canon_sha256":"e7dda51325b02b529fe2d517d7d20b511fa2f95ba08b7275753990941f84bf54"},"schema_version":"1.0","source":{"id":"1206.5926","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1206.5926","created_at":"2026-05-18T03:27:11Z"},{"alias_kind":"arxiv_version","alias_value":"1206.5926v2","created_at":"2026-05-18T03:27:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.5926","created_at":"2026-05-18T03:27:11Z"},{"alias_kind":"pith_short_12","alias_value":"XPQRMLEBCGIN","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_16","alias_value":"XPQRMLEBCGINZ7GU","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_8","alias_value":"XPQRMLEB","created_at":"2026-05-18T12:27:27Z"}],"graph_snapshots":[{"event_id":"sha256:cc9b30f5a4bd193e28eb14dbd1f308b9c756572b379b38157e4ced41156c0baf","target":"graph","created_at":"2026-05-18T03:27:11Z","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 polynomial D(G,x) of a graph G is the generating function of its dominating sets. We prove that D(G,x) satisfies a wide range of reduction formulas. We show linear recurrence relations for D(G,x) for arbitrary graphs and for various special cases. We give splitting formulas for D(G,x) based on articulation vertices, and more generally, on splitting sets of vertices.","authors_text":"Frank Simon, James Preen, Martin Trinks, Peter Tittmann, Tomer Kotek","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-06-26T08:58:50Z","title":"Recurrence relations and splitting formulas for the domination polynomial"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.5926","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":"sha256:1492659303c1f4fc8e499b0e3fea4d78f702863a4fdd83512e9e73383e241718","target":"record","created_at":"2026-05-18T03:27:11Z","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":"84f71f3a687988071fd1c8fe513558be6f44a84e0ac080b68352c04c86705171","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-06-26T08:58:50Z","title_canon_sha256":"e7dda51325b02b529fe2d517d7d20b511fa2f95ba08b7275753990941f84bf54"},"schema_version":"1.0","source":{"id":"1206.5926","kind":"arxiv","version":2}},"canonical_sha256":"bbe1162c811190dcfcd4cc336480da5774a0f3eb61f0201ae4a8e2e0b99b8150","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bbe1162c811190dcfcd4cc336480da5774a0f3eb61f0201ae4a8e2e0b99b8150","first_computed_at":"2026-05-18T03:27:11.442616Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:27:11.442616Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"l489/IWE8M6U7bfZ9uAonVUXuCIr1EWRQMQQqseB4OmKZkSQBAmj7jCXNtEcOMNMN08EXnjN/hIyIZSx2ivXBw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:27:11.443199Z","signed_message":"canonical_sha256_bytes"},"source_id":"1206.5926","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1492659303c1f4fc8e499b0e3fea4d78f702863a4fdd83512e9e73383e241718","sha256:cc9b30f5a4bd193e28eb14dbd1f308b9c756572b379b38157e4ced41156c0baf"],"state_sha256":"6b2f1a8e7e0ae80b89905a76de2a11399a43fb7c20ec838ceec5c22bd2b8b8fb"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Tfm/Rkcai3OnSrFJNEEDooMlJVSvZaziQNOT1eGvKhJngzYxbjnPWyfv9lQwyCWYCwgsDLdzGwMKUumnFD0aBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T01:41:53.147612Z","bundle_sha256":"fd0b5beed2aae1b148e59623e595479df65281517d83279e3a16186408220e6c"}}