{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:XVMLQG3SWVIIMGXWVOJ2R6NIGH","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":"6e6182ce2162a809beab75bb54fe1b7edbadadebb03f76742a2074a93832c335","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-08-11T17:07:16Z","title_canon_sha256":"d1129ca1ebd57101f47a5ba9cf79d45e7831ae3fdc9451efe69cbdaef472ee0c"},"schema_version":"1.0","source":{"id":"1608.03538","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1608.03538","created_at":"2026-05-18T00:24:31Z"},{"alias_kind":"arxiv_version","alias_value":"1608.03538v1","created_at":"2026-05-18T00:24:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.03538","created_at":"2026-05-18T00:24:31Z"},{"alias_kind":"pith_short_12","alias_value":"XVMLQG3SWVII","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_16","alias_value":"XVMLQG3SWVIIMGXW","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_8","alias_value":"XVMLQG3S","created_at":"2026-05-18T12:30:51Z"}],"graph_snapshots":[{"event_id":"sha256:9b8e3a538467af5475998e07f6932de575eb212bab06435d1fca6cc8ade9a751","target":"graph","created_at":"2026-05-18T00:24:31Z","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":"We discuss a partial normalisation of a finite graph of finite groups $(\\Gamma(-), X)$ which leaves invariant the fundamental group. In conjunction with an easy graph-theoretic result, this provides a flexible and rather useful tool in the study of finitely generated virtually free groups. Applications discussed here include (i) an important inequality for the number of edges in a Stallings decomposition $\\Gamma \\cong \\pi_1(\\Gamma(-), X)$ of a finitely generated virtually free group, (ii) the proof of equivalence of a number of conditions for such a group to be `large', as well as (iii) the cl","authors_text":"Christian Krattenthaler (Universit\\\"at Wien), Thomas W. M\\\"uller (Queen Mary, University of London), Westfield College","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-08-11T17:07:16Z","title":"Normalising graphs of groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.03538","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:ee7f43fbf1c15d8c1ade63ad5d1925556e1e89317cdbc3768607f5d004b8f9ff","target":"record","created_at":"2026-05-18T00:24:31Z","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":"6e6182ce2162a809beab75bb54fe1b7edbadadebb03f76742a2074a93832c335","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-08-11T17:07:16Z","title_canon_sha256":"d1129ca1ebd57101f47a5ba9cf79d45e7831ae3fdc9451efe69cbdaef472ee0c"},"schema_version":"1.0","source":{"id":"1608.03538","kind":"arxiv","version":1}},"canonical_sha256":"bd58b81b72b550861af6ab93a8f9a831f2b8f88c0a6cd63a064bfbeb6c44abb5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bd58b81b72b550861af6ab93a8f9a831f2b8f88c0a6cd63a064bfbeb6c44abb5","first_computed_at":"2026-05-18T00:24:31.758118Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:24:31.758118Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"AmBKbpeeU5BWY9Ie6lQ8kAft71FKC+XnYlLQrK0faOEXYuVORVPQ6+qG0JvVDF1KgEEiIo8Jj08BddF6CaGEBw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:24:31.758742Z","signed_message":"canonical_sha256_bytes"},"source_id":"1608.03538","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ee7f43fbf1c15d8c1ade63ad5d1925556e1e89317cdbc3768607f5d004b8f9ff","sha256:9b8e3a538467af5475998e07f6932de575eb212bab06435d1fca6cc8ade9a751"],"state_sha256":"b7ceab2b598d47cacbf800be2f08ff354c6fbf8f0fbc02a6cef6c85d2583403e"}