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Let $T$ be a digraph with $t$ vertices $u_1,\\dots , u_t$ and let $H_1,\\dots H_t$ be digraphs such that $H_i$ has vertices $u_{i,j_i},\\ 1\\le j_i\\le n_i.$ Then the composition $Q=T[H_1,\\dots , H_t]$ is a digraph with vertex set $\\{u_{i,j_i}\\mid 1\\le i\\le t, 1\\le j_i\\le n_i\\}$ and arc set $$A(Q)=\\cup^t_{i=1}A(H_i)\\cup \\{u_{ij_i}u_{pq_p}\\mid u_iu_p\\in A(T), 1\\le j_i\\le n_i, 1\\le q_p\\le n_p\\}.$$\n  When $T$ is arbitrary, we obtain the following result: every strong digraph com"},"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":"1906.08052","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2019-06-19T12:18:21Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"d7c5fa3487b7fdd00b45c75ee98a9637a060e44efeba41ff012a784874a3d8e7","abstract_canon_sha256":"4a1653d1af16e2bf6a68a47e7f0edd4c740bd07db6d35722a4657c6b53ea9094"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:42:54.753300Z","signature_b64":"0kroxqYB2XCAWQDL0L6pSeURdbHRt4ZwuXb9KSEo1UMs66WzIZsaBhgTbWPWYBLjItywvW+viPBTHxOfsCn7Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f0de5935157b53af6c8fcf9e4bebc6aaddc9f54ee23b85d7ea08897534f9d39f","last_reissued_at":"2026-05-17T23:42:54.752709Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:42:54.752709Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Arc-disjoint in- and out-branchings rooted at the same vertex in compositions of digraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.DM","authors_text":"Gregory Gutin, Yuefang Sun","submitted_at":"2019-06-19T12:18:21Z","abstract_excerpt":"A digraph $D=(V, A)$ has a good pair at a vertex $r$ if $D$ has a pair of arc-disjoint in- and out-branchings rooted at $r$. Let $T$ be a digraph with $t$ vertices $u_1,\\dots , u_t$ and let $H_1,\\dots H_t$ be digraphs such that $H_i$ has vertices $u_{i,j_i},\\ 1\\le j_i\\le n_i.$ Then the composition $Q=T[H_1,\\dots , H_t]$ is a digraph with vertex set $\\{u_{i,j_i}\\mid 1\\le i\\le t, 1\\le j_i\\le n_i\\}$ and arc set $$A(Q)=\\cup^t_{i=1}A(H_i)\\cup \\{u_{ij_i}u_{pq_p}\\mid u_iu_p\\in A(T), 1\\le j_i\\le n_i, 1\\le q_p\\le n_p\\}.$$\n  When $T$ is arbitrary, we obtain the following result: every strong digraph com"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.08052","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":"1906.08052","created_at":"2026-05-17T23:42:54.752806+00:00"},{"alias_kind":"arxiv_version","alias_value":"1906.08052v1","created_at":"2026-05-17T23:42:54.752806+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1906.08052","created_at":"2026-05-17T23:42:54.752806+00:00"},{"alias_kind":"pith_short_12","alias_value":"6DPFSNIVPNJ2","created_at":"2026-05-18T12:33:10.108867+00:00"},{"alias_kind":"pith_short_16","alias_value":"6DPFSNIVPNJ263EP","created_at":"2026-05-18T12:33:10.108867+00:00"},{"alias_kind":"pith_short_8","alias_value":"6DPFSNIV","created_at":"2026-05-18T12:33:10.108867+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/6DPFSNIVPNJ263EPZ6PEX26GVL","json":"https://pith.science/pith/6DPFSNIVPNJ263EPZ6PEX26GVL.json","graph_json":"https://pith.science/api/pith-number/6DPFSNIVPNJ263EPZ6PEX26GVL/graph.json","events_json":"https://pith.science/api/pith-number/6DPFSNIVPNJ263EPZ6PEX26GVL/events.json","paper":"https://pith.science/paper/6DPFSNIV"},"agent_actions":{"view_html":"https://pith.science/pith/6DPFSNIVPNJ263EPZ6PEX26GVL","download_json":"https://pith.science/pith/6DPFSNIVPNJ263EPZ6PEX26GVL.json","view_paper":"https://pith.science/paper/6DPFSNIV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1906.08052&json=true","fetch_graph":"https://pith.science/api/pith-number/6DPFSNIVPNJ263EPZ6PEX26GVL/graph.json","fetch_events":"https://pith.science/api/pith-number/6DPFSNIVPNJ263EPZ6PEX26GVL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6DPFSNIVPNJ263EPZ6PEX26GVL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6DPFSNIVPNJ263EPZ6PEX26GVL/action/storage_attestation","attest_author":"https://pith.science/pith/6DPFSNIVPNJ263EPZ6PEX26GVL/action/author_attestation","sign_citation":"https://pith.science/pith/6DPFSNIVPNJ263EPZ6PEX26GVL/action/citation_signature","submit_replication":"https://pith.science/pith/6DPFSNIVPNJ263EPZ6PEX26GVL/action/replication_record"}},"created_at":"2026-05-17T23:42:54.752806+00:00","updated_at":"2026-05-17T23:42:54.752806+00:00"}