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We settle the complexity of the following two problems:\n  1) Given a 2-regular digraph $D$, decide if it contains two arc-disjoint branchings B^+_u, B^-_v.\n  2) Given a 2-regular digraph D, decide if it contains an out-branching B^+_u such that D remains connected after removing the arcs of B^+_u.\n  Both problems are NP-complete for general digraphs. We prove that the first proble"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1203.4705","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-03-21T11:15:51Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"04289276c76240f5bdf64686726b44cd8b086ff7c3b297c07798a29d48bb4281","abstract_canon_sha256":"d2222c38f3714bba8927660919e0e8f109f62c030c1e8a5604a5f438e29c70a5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:59:37.405636Z","signature_b64":"Hfp23ctZh6LjfwDvQcYzf4SN3NyHJp3C/z/6/Z0EDpax0+SlazFsPSJYRN0R83VBSCZVa6iVtfDTGCeyHOhmCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2295b37e6e01a180377bb2eb26aa88ef09208d15d1972c242ecb136a071c0d28","last_reissued_at":"2026-05-18T03:59:37.404156Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:59:37.404156Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Arc-Disjoint Paths and Trees in 2-Regular Digraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"J{\\o}rgen Bang-Jensen, Sven Simonsen","submitted_at":"2012-03-21T11:15:51Z","abstract_excerpt":"An out-(in-)branching B_s^+ (B_s^-) rooted at s in a digraph D is a connected spanning subdigraph of D in which every vertex x != s has precisely one arc entering (leaving) it and s has no arcs entering (leaving) it. We settle the complexity of the following two problems:\n  1) Given a 2-regular digraph $D$, decide if it contains two arc-disjoint branchings B^+_u, B^-_v.\n  2) Given a 2-regular digraph D, decide if it contains an out-branching B^+_u such that D remains connected after removing the arcs of B^+_u.\n  Both problems are NP-complete for general digraphs. We prove that the first proble"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.4705","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1203.4705","created_at":"2026-05-18T03:59:37.404276+00:00"},{"alias_kind":"arxiv_version","alias_value":"1203.4705v1","created_at":"2026-05-18T03:59:37.404276+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1203.4705","created_at":"2026-05-18T03:59:37.404276+00:00"},{"alias_kind":"pith_short_12","alias_value":"EKK3G7TOAGQY","created_at":"2026-05-18T12:27:04.183437+00:00"},{"alias_kind":"pith_short_16","alias_value":"EKK3G7TOAGQYAN33","created_at":"2026-05-18T12:27:04.183437+00:00"},{"alias_kind":"pith_short_8","alias_value":"EKK3G7TO","created_at":"2026-05-18T12:27:04.183437+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/EKK3G7TOAGQYAN33WLVSNKUI54","json":"https://pith.science/pith/EKK3G7TOAGQYAN33WLVSNKUI54.json","graph_json":"https://pith.science/api/pith-number/EKK3G7TOAGQYAN33WLVSNKUI54/graph.json","events_json":"https://pith.science/api/pith-number/EKK3G7TOAGQYAN33WLVSNKUI54/events.json","paper":"https://pith.science/paper/EKK3G7TO"},"agent_actions":{"view_html":"https://pith.science/pith/EKK3G7TOAGQYAN33WLVSNKUI54","download_json":"https://pith.science/pith/EKK3G7TOAGQYAN33WLVSNKUI54.json","view_paper":"https://pith.science/paper/EKK3G7TO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1203.4705&json=true","fetch_graph":"https://pith.science/api/pith-number/EKK3G7TOAGQYAN33WLVSNKUI54/graph.json","fetch_events":"https://pith.science/api/pith-number/EKK3G7TOAGQYAN33WLVSNKUI54/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/EKK3G7TOAGQYAN33WLVSNKUI54/action/timestamp_anchor","attest_storage":"https://pith.science/pith/EKK3G7TOAGQYAN33WLVSNKUI54/action/storage_attestation","attest_author":"https://pith.science/pith/EKK3G7TOAGQYAN33WLVSNKUI54/action/author_attestation","sign_citation":"https://pith.science/pith/EKK3G7TOAGQYAN33WLVSNKUI54/action/citation_signature","submit_replication":"https://pith.science/pith/EKK3G7TOAGQYAN33WLVSNKUI54/action/replication_record"}},"created_at":"2026-05-18T03:59:37.404276+00:00","updated_at":"2026-05-18T03:59:37.404276+00:00"}