{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:YJDS3OTCIUUHUCHN6PPVNYLKEO","short_pith_number":"pith:YJDS3OTC","canonical_record":{"source":{"id":"1108.3588","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-08-17T22:16:26Z","cross_cats_sorted":[],"title_canon_sha256":"56c65f08a633e18b30dfbe179447ed4ca879b090bf1780730a7638644af0bec8","abstract_canon_sha256":"37ef4b7481666e8b40d6045af99121dc0f3e9f92ca474f5ad8ca494ab236cb7d"},"schema_version":"1.0"},"canonical_sha256":"c2472dba6245287a08edf3df56e16a23bcaa3377f5d35940e900e77a85003c93","source":{"kind":"arxiv","id":"1108.3588","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1108.3588","created_at":"2026-05-18T04:15:15Z"},{"alias_kind":"arxiv_version","alias_value":"1108.3588v1","created_at":"2026-05-18T04:15:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1108.3588","created_at":"2026-05-18T04:15:15Z"},{"alias_kind":"pith_short_12","alias_value":"YJDS3OTCIUUH","created_at":"2026-05-18T12:26:47Z"},{"alias_kind":"pith_short_16","alias_value":"YJDS3OTCIUUHUCHN","created_at":"2026-05-18T12:26:47Z"},{"alias_kind":"pith_short_8","alias_value":"YJDS3OTC","created_at":"2026-05-18T12:26:47Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:YJDS3OTCIUUHUCHN6PPVNYLKEO","target":"record","payload":{"canonical_record":{"source":{"id":"1108.3588","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-08-17T22:16:26Z","cross_cats_sorted":[],"title_canon_sha256":"56c65f08a633e18b30dfbe179447ed4ca879b090bf1780730a7638644af0bec8","abstract_canon_sha256":"37ef4b7481666e8b40d6045af99121dc0f3e9f92ca474f5ad8ca494ab236cb7d"},"schema_version":"1.0"},"canonical_sha256":"c2472dba6245287a08edf3df56e16a23bcaa3377f5d35940e900e77a85003c93","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:15:15.894544Z","signature_b64":"xH6NDglQvoDytBIt0T6y6Qsnn3dhFjlBGJ6+fk2onH/tDy/g29qymWb1CTDw8Vv56Ck8hAlaDMllyskQIZLhBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c2472dba6245287a08edf3df56e16a23bcaa3377f5d35940e900e77a85003c93","last_reissued_at":"2026-05-18T04:15:15.893809Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:15:15.893809Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1108.3588","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":"2026-05-18T04:15:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4Kwnn00fw8Z/iKygdAfMIsrNtq+ebzXLKUz7qicXGs/hsrjdjJQowyS+ecDi4/De/Z2vKdVEf78GqtOor4D9Bg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T07:13:30.730488Z"},"content_sha256":"179b54497d748a226b68851c6057728253135daf458c870e80eaa089cab53a98","schema_version":"1.0","event_id":"sha256:179b54497d748a226b68851c6057728253135daf458c870e80eaa089cab53a98"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:YJDS3OTCIUUHUCHN6PPVNYLKEO","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Graph Invertibility","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Cam McLeman, Erin McNicholas","submitted_at":"2011-08-17T22:16:26Z","abstract_excerpt":"Extending the work of Godsil and others, we investigate the notion of the inverse of a graph (specifically, of bipartite graphs with a unique perfect matching). We provide a concise necessary and sufficient condition for the invertibility of such graphs and generalize the notion of invertibility to multigraphs. We examine the question of whether there exists a \"litmus subgraph\" whose bipartiteness determines invertibility. As an application of our invertibility criteria, we quickly describe all invertible unicyclic graphs. Finally, we describe a general combinatorial procedure for iteratively "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.3588","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":"2026-05-18T04:15:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"U8Iz/jSwlFCkxYF5yuXmyIEkF8RCgovoDRlIooF+0BWZaecVbKI3o+U6QSBxAp7RusSWoPOkhjSvfj2Zbk4aDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T07:13:30.730858Z"},"content_sha256":"24a841f437817c6a4f78ea1e37dea6e2e5dd685daae768be7a5673d530f90b0d","schema_version":"1.0","event_id":"sha256:24a841f437817c6a4f78ea1e37dea6e2e5dd685daae768be7a5673d530f90b0d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/YJDS3OTCIUUHUCHN6PPVNYLKEO/bundle.json","state_url":"https://pith.science/pith/YJDS3OTCIUUHUCHN6PPVNYLKEO/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/YJDS3OTCIUUHUCHN6PPVNYLKEO/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":"2026-06-04T07:13:30Z","links":{"resolver":"https://pith.science/pith/YJDS3OTCIUUHUCHN6PPVNYLKEO","bundle":"https://pith.science/pith/YJDS3OTCIUUHUCHN6PPVNYLKEO/bundle.json","state":"https://pith.science/pith/YJDS3OTCIUUHUCHN6PPVNYLKEO/state.json","well_known_bundle":"https://pith.science/.well-known/pith/YJDS3OTCIUUHUCHN6PPVNYLKEO/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:YJDS3OTCIUUHUCHN6PPVNYLKEO","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":"37ef4b7481666e8b40d6045af99121dc0f3e9f92ca474f5ad8ca494ab236cb7d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-08-17T22:16:26Z","title_canon_sha256":"56c65f08a633e18b30dfbe179447ed4ca879b090bf1780730a7638644af0bec8"},"schema_version":"1.0","source":{"id":"1108.3588","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1108.3588","created_at":"2026-05-18T04:15:15Z"},{"alias_kind":"arxiv_version","alias_value":"1108.3588v1","created_at":"2026-05-18T04:15:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1108.3588","created_at":"2026-05-18T04:15:15Z"},{"alias_kind":"pith_short_12","alias_value":"YJDS3OTCIUUH","created_at":"2026-05-18T12:26:47Z"},{"alias_kind":"pith_short_16","alias_value":"YJDS3OTCIUUHUCHN","created_at":"2026-05-18T12:26:47Z"},{"alias_kind":"pith_short_8","alias_value":"YJDS3OTC","created_at":"2026-05-18T12:26:47Z"}],"graph_snapshots":[{"event_id":"sha256:24a841f437817c6a4f78ea1e37dea6e2e5dd685daae768be7a5673d530f90b0d","target":"graph","created_at":"2026-05-18T04:15:15Z","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":"Extending the work of Godsil and others, we investigate the notion of the inverse of a graph (specifically, of bipartite graphs with a unique perfect matching). We provide a concise necessary and sufficient condition for the invertibility of such graphs and generalize the notion of invertibility to multigraphs. We examine the question of whether there exists a \"litmus subgraph\" whose bipartiteness determines invertibility. As an application of our invertibility criteria, we quickly describe all invertible unicyclic graphs. Finally, we describe a general combinatorial procedure for iteratively ","authors_text":"Cam McLeman, Erin McNicholas","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-08-17T22:16:26Z","title":"Graph Invertibility"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.3588","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":"sha256:179b54497d748a226b68851c6057728253135daf458c870e80eaa089cab53a98","target":"record","created_at":"2026-05-18T04:15:15Z","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":"37ef4b7481666e8b40d6045af99121dc0f3e9f92ca474f5ad8ca494ab236cb7d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-08-17T22:16:26Z","title_canon_sha256":"56c65f08a633e18b30dfbe179447ed4ca879b090bf1780730a7638644af0bec8"},"schema_version":"1.0","source":{"id":"1108.3588","kind":"arxiv","version":1}},"canonical_sha256":"c2472dba6245287a08edf3df56e16a23bcaa3377f5d35940e900e77a85003c93","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c2472dba6245287a08edf3df56e16a23bcaa3377f5d35940e900e77a85003c93","first_computed_at":"2026-05-18T04:15:15.893809Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:15:15.893809Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"xH6NDglQvoDytBIt0T6y6Qsnn3dhFjlBGJ6+fk2onH/tDy/g29qymWb1CTDw8Vv56Ck8hAlaDMllyskQIZLhBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:15:15.894544Z","signed_message":"canonical_sha256_bytes"},"source_id":"1108.3588","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:179b54497d748a226b68851c6057728253135daf458c870e80eaa089cab53a98","sha256:24a841f437817c6a4f78ea1e37dea6e2e5dd685daae768be7a5673d530f90b0d"],"state_sha256":"2bb9e230664fe4ecd85da41d1006c9b28502b730c2f8acc1f623a3f6d517b8e9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mRZrc4mxm4TmlvtADzVXaeMEAUZpGEAA5G4puiZ9r5G+bBpWNuzHiDFTNMFFcXK9giC3L7bjT2+izMpvUSDeAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-04T07:13:30.733276Z","bundle_sha256":"d7edd8a1763a0997db565e3fbb173a672c53d5194e45b80fa86fdd0ff712c054"}}