{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2007:WU2U3T563FDFLOL2XU7MP7EZHA","short_pith_number":"pith:WU2U3T56","canonical_record":{"source":{"id":"0712.2908","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.CO","submitted_at":"2007-12-18T09:46:07Z","cross_cats_sorted":[],"title_canon_sha256":"3dc5ed7f1fce3c3fa08e5205285f6bf8a022fa960151dbeef516348121aa9105","abstract_canon_sha256":"18ff4e2bc559aff301a48dc64d296ee0c68a113fe4428606f56b26e30e2fcb91"},"schema_version":"1.0"},"canonical_sha256":"b5354dcfbed94655b97abd3ec7fc9938089b1c34046411a396524d6fad1e5d47","source":{"kind":"arxiv","id":"0712.2908","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0712.2908","created_at":"2026-05-18T04:09:04Z"},{"alias_kind":"arxiv_version","alias_value":"0712.2908v2","created_at":"2026-05-18T04:09:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0712.2908","created_at":"2026-05-18T04:09:04Z"},{"alias_kind":"pith_short_12","alias_value":"WU2U3T563FDF","created_at":"2026-05-18T12:25:56Z"},{"alias_kind":"pith_short_16","alias_value":"WU2U3T563FDFLOL2","created_at":"2026-05-18T12:25:56Z"},{"alias_kind":"pith_short_8","alias_value":"WU2U3T56","created_at":"2026-05-18T12:25:56Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2007:WU2U3T563FDFLOL2XU7MP7EZHA","target":"record","payload":{"canonical_record":{"source":{"id":"0712.2908","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.CO","submitted_at":"2007-12-18T09:46:07Z","cross_cats_sorted":[],"title_canon_sha256":"3dc5ed7f1fce3c3fa08e5205285f6bf8a022fa960151dbeef516348121aa9105","abstract_canon_sha256":"18ff4e2bc559aff301a48dc64d296ee0c68a113fe4428606f56b26e30e2fcb91"},"schema_version":"1.0"},"canonical_sha256":"b5354dcfbed94655b97abd3ec7fc9938089b1c34046411a396524d6fad1e5d47","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:09:04.270177Z","signature_b64":"MhTpfFHRJQdnWn6KZxghU7fwMwCbOERUm392UoL367HOo+BT3dtu66qQV0eZwEWh8iFuaJA2mGkWvkYuS9kbCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b5354dcfbed94655b97abd3ec7fc9938089b1c34046411a396524d6fad1e5d47","last_reissued_at":"2026-05-18T04:09:04.269660Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:09:04.269660Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0712.2908","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":"2026-05-18T04:09:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"EmgGASNPD+IDBzBaKUgaxlG3aomAWa3g/EpZwryQYCDF1cOQOwjpY3j9roAwrZZebJk+n0vJZEhO7yPT4zgXAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T20:47:18.308693Z"},"content_sha256":"53d806693c61896d279494f5ed690b71afcd33c9a65956a61f25fc26b1245ce9","schema_version":"1.0","event_id":"sha256:53d806693c61896d279494f5ed690b71afcd33c9a65956a61f25fc26b1245ce9"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2007:WU2U3T563FDFLOL2XU7MP7EZHA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Cops & Robber game on series-parallel graphs","license":"","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dirk Oliver Theis","submitted_at":"2007-12-18T09:46:07Z","abstract_excerpt":"The Cops and Robber game is played on undirected finite graphs. $k$ cops and one robber are positioned on vertices and take turn in moving along edges. The cops win if, after a move, a cop and the robber are on the same vertex. A graph is called $k$-copwin, if the cops have a winning strategy.\n  It is known that planar graphs are 3-copwin (Aigner & Fromme, 1984) and that outerplanar graphs are 2-copwin (Clarke, 2002). In this short note, we prove that series-parallel (i.e., graphs with no $K_4$ minor) graphs are 2-copwin.\n  It is a well-known trick in the literature of cops & robber games to d"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0712.2908","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":"2026-05-18T04:09:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Qjs6jtmpFoSJPI6Oo1VjwEw3cBWSvSaNQXfPql4/u3diQzkwS3G4d+eAb69gdsI1ZGVRaY/+bl7q0RI005gCAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T20:47:18.309040Z"},"content_sha256":"9211f09ea0e56ada1c066b7fdf0e62e0fae258c40ced0379d6f982b136449f2a","schema_version":"1.0","event_id":"sha256:9211f09ea0e56ada1c066b7fdf0e62e0fae258c40ced0379d6f982b136449f2a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WU2U3T563FDFLOL2XU7MP7EZHA/bundle.json","state_url":"https://pith.science/pith/WU2U3T563FDFLOL2XU7MP7EZHA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WU2U3T563FDFLOL2XU7MP7EZHA/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":"2026-06-02T20:47:18Z","links":{"resolver":"https://pith.science/pith/WU2U3T563FDFLOL2XU7MP7EZHA","bundle":"https://pith.science/pith/WU2U3T563FDFLOL2XU7MP7EZHA/bundle.json","state":"https://pith.science/pith/WU2U3T563FDFLOL2XU7MP7EZHA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WU2U3T563FDFLOL2XU7MP7EZHA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2007:WU2U3T563FDFLOL2XU7MP7EZHA","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":"18ff4e2bc559aff301a48dc64d296ee0c68a113fe4428606f56b26e30e2fcb91","cross_cats_sorted":[],"license":"","primary_cat":"math.CO","submitted_at":"2007-12-18T09:46:07Z","title_canon_sha256":"3dc5ed7f1fce3c3fa08e5205285f6bf8a022fa960151dbeef516348121aa9105"},"schema_version":"1.0","source":{"id":"0712.2908","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0712.2908","created_at":"2026-05-18T04:09:04Z"},{"alias_kind":"arxiv_version","alias_value":"0712.2908v2","created_at":"2026-05-18T04:09:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0712.2908","created_at":"2026-05-18T04:09:04Z"},{"alias_kind":"pith_short_12","alias_value":"WU2U3T563FDF","created_at":"2026-05-18T12:25:56Z"},{"alias_kind":"pith_short_16","alias_value":"WU2U3T563FDFLOL2","created_at":"2026-05-18T12:25:56Z"},{"alias_kind":"pith_short_8","alias_value":"WU2U3T56","created_at":"2026-05-18T12:25:56Z"}],"graph_snapshots":[{"event_id":"sha256:9211f09ea0e56ada1c066b7fdf0e62e0fae258c40ced0379d6f982b136449f2a","target":"graph","created_at":"2026-05-18T04:09:04Z","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 Cops and Robber game is played on undirected finite graphs. $k$ cops and one robber are positioned on vertices and take turn in moving along edges. The cops win if, after a move, a cop and the robber are on the same vertex. A graph is called $k$-copwin, if the cops have a winning strategy.\n  It is known that planar graphs are 3-copwin (Aigner & Fromme, 1984) and that outerplanar graphs are 2-copwin (Clarke, 2002). In this short note, we prove that series-parallel (i.e., graphs with no $K_4$ minor) graphs are 2-copwin.\n  It is a well-known trick in the literature of cops & robber games to d","authors_text":"Dirk Oliver Theis","cross_cats":[],"headline":"","license":"","primary_cat":"math.CO","submitted_at":"2007-12-18T09:46:07Z","title":"The Cops & Robber game on series-parallel graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0712.2908","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":"sha256:53d806693c61896d279494f5ed690b71afcd33c9a65956a61f25fc26b1245ce9","target":"record","created_at":"2026-05-18T04:09:04Z","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":"18ff4e2bc559aff301a48dc64d296ee0c68a113fe4428606f56b26e30e2fcb91","cross_cats_sorted":[],"license":"","primary_cat":"math.CO","submitted_at":"2007-12-18T09:46:07Z","title_canon_sha256":"3dc5ed7f1fce3c3fa08e5205285f6bf8a022fa960151dbeef516348121aa9105"},"schema_version":"1.0","source":{"id":"0712.2908","kind":"arxiv","version":2}},"canonical_sha256":"b5354dcfbed94655b97abd3ec7fc9938089b1c34046411a396524d6fad1e5d47","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b5354dcfbed94655b97abd3ec7fc9938089b1c34046411a396524d6fad1e5d47","first_computed_at":"2026-05-18T04:09:04.269660Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:09:04.269660Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"MhTpfFHRJQdnWn6KZxghU7fwMwCbOERUm392UoL367HOo+BT3dtu66qQV0eZwEWh8iFuaJA2mGkWvkYuS9kbCA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:09:04.270177Z","signed_message":"canonical_sha256_bytes"},"source_id":"0712.2908","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:53d806693c61896d279494f5ed690b71afcd33c9a65956a61f25fc26b1245ce9","sha256:9211f09ea0e56ada1c066b7fdf0e62e0fae258c40ced0379d6f982b136449f2a"],"state_sha256":"ebb112682656986516c39fcf719b898fb00f32fe55449d81ae9ca5e0754f1ad2"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UMGh+jjBdGxChyItyg/1dSuos71VJdlRjSgl1cJV25fTYrql67HVjDpCp4tv++zK4VWmAN7w4AyWtL/oSR2dCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T20:47:18.311023Z","bundle_sha256":"9ce878add9251ad9e51cab5826dd85393de02a0d481818b04e5bce590dfcae28"}}