{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:CO7YXUACKP7AQDSKQSWJHD5BOH","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":"cab6e4b1a1ee63e76b1b7701c42b7c0b7611dfe0181478e747186f6050ffae11","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-06-21T17:08:03Z","title_canon_sha256":"d5d043784ee50b18636a3a8b2b49292dc5d0d1503d9d391e9b067d33d8e51bda"},"schema_version":"1.0","source":{"id":"1606.06663","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1606.06663","created_at":"2026-05-18T00:53:50Z"},{"alias_kind":"arxiv_version","alias_value":"1606.06663v2","created_at":"2026-05-18T00:53:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.06663","created_at":"2026-05-18T00:53:50Z"},{"alias_kind":"pith_short_12","alias_value":"CO7YXUACKP7A","created_at":"2026-05-18T12:30:09Z"},{"alias_kind":"pith_short_16","alias_value":"CO7YXUACKP7AQDSK","created_at":"2026-05-18T12:30:09Z"},{"alias_kind":"pith_short_8","alias_value":"CO7YXUAC","created_at":"2026-05-18T12:30:09Z"}],"graph_snapshots":[{"event_id":"sha256:b1b684cfd3ab13e9690553646c5754129a7f876b3a4605d530aa633e06b95972","target":"graph","created_at":"2026-05-18T00:53:50Z","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":"A tree with at most k leaves is called k-ended tree, and a tree with exactly k leaves is called k-end tree, where a leaf is a vertex of degree one. Contraction of a graph G along the edge e means deleting the edge e and identifying its end vertices and deleting all edges between every two vertex except one edge to gain again a simple graph and is denoted bye G/e. In this paper we prove some theorems related to a graph and its contraction. For example we prove the following theorem. If G is a connected graph that has a spanning k-end tree and |V (G)| > K + 1 then there exist an edge e such G/e ","authors_text":"Hamed Ghasemian Zoeram","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-06-21T17:08:03Z","title":"Contraction of graphs and spanning k-end trees"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.06663","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":"sha256:49d268493e85895fcd0792cc4dc7bf5ba253df0e9171c39e991e9218320568b8","target":"record","created_at":"2026-05-18T00:53:50Z","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":"cab6e4b1a1ee63e76b1b7701c42b7c0b7611dfe0181478e747186f6050ffae11","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-06-21T17:08:03Z","title_canon_sha256":"d5d043784ee50b18636a3a8b2b49292dc5d0d1503d9d391e9b067d33d8e51bda"},"schema_version":"1.0","source":{"id":"1606.06663","kind":"arxiv","version":2}},"canonical_sha256":"13bf8bd00253fe080e4a84ac938fa171d65be218dacf085de2aad25b06cdbb7a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"13bf8bd00253fe080e4a84ac938fa171d65be218dacf085de2aad25b06cdbb7a","first_computed_at":"2026-05-18T00:53:50.882194Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:53:50.882194Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"mk4ZM4qwtFq741QwOnOcnawT7YGhicwwQe2vJ4cLuM/duSJ1ON5AVMxvKaVZjQodOLLspVWYDW59ymQkToo6Ag==","signature_status":"signed_v1","signed_at":"2026-05-18T00:53:50.882603Z","signed_message":"canonical_sha256_bytes"},"source_id":"1606.06663","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:49d268493e85895fcd0792cc4dc7bf5ba253df0e9171c39e991e9218320568b8","sha256:b1b684cfd3ab13e9690553646c5754129a7f876b3a4605d530aa633e06b95972"],"state_sha256":"e29d7c462fd97fa2d1593a5e844c0d2ef06abca5f55fa3844c10c4e32b68deb0"}