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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 $\\cup_{i=1}^t V(H_i)=\\{u_{i,j_i}\\mid 1\\le i\\le t, 1\\le j_i\\le n_i\\}$ and arc set \\[ \\left(\\cup^t_{i=1}A(H_i) \\right) \\cup \\left( \\cup_{u_iu_p\\in A(T)} \\{u_{i"},"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":"1903.12225","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2019-03-28T19:02:20Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"63e61e9612d5cac5267f0bb0b578444f134602a110453ae3d7b1c4c958c09c1a","abstract_canon_sha256":"0b3e94f33a75c353f7df39d621406dc74e6d2c38353fc6b06221f56801f6684e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:49:54.550555Z","signature_b64":"btPk3S8wB4aQTY3KQKCVGh8CNKJGNBtTNixOFt6qUoEgLV5f/UAN2HhjVT9HG15qdPoLpLvDY6fALw2weiGMBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e9c9a16daf986b6bede67acd16245a36ca6acb52e2b188a1a7ef0ab886bb3317","last_reissued_at":"2026-05-17T23:49:54.549993Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:49:54.549993Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Arc-disjoint Strong Spanning Subdigraphs of Semicomplete Compositions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.DM","authors_text":"Anders Yeo, Gregory Gutin, Joergen Bang-Jensen","submitted_at":"2019-03-28T19:02:20Z","abstract_excerpt":"A strong arc decomposition of a digraph $D=(V,A)$ is a decomposition of its arc set $A$ into two disjoint subsets $A_1$ and $A_2$ such that both of the spanning subdigraphs $D_1=(V,A_1)$ and $D_2=(V,A_2)$ are strong. 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 $\\cup_{i=1}^t V(H_i)=\\{u_{i,j_i}\\mid 1\\le i\\le t, 1\\le j_i\\le n_i\\}$ and arc set \\[ \\left(\\cup^t_{i=1}A(H_i) \\right) \\cup \\left( \\cup_{u_iu_p\\in A(T)} \\{u_{i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1903.12225","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":"1903.12225","created_at":"2026-05-17T23:49:54.550164+00:00"},{"alias_kind":"arxiv_version","alias_value":"1903.12225v1","created_at":"2026-05-17T23:49:54.550164+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1903.12225","created_at":"2026-05-17T23:49:54.550164+00:00"},{"alias_kind":"pith_short_12","alias_value":"5HE2C3NPTBVW","created_at":"2026-05-18T12:33:10.108867+00:00"},{"alias_kind":"pith_short_16","alias_value":"5HE2C3NPTBVWX3PG","created_at":"2026-05-18T12:33:10.108867+00:00"},{"alias_kind":"pith_short_8","alias_value":"5HE2C3NP","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/5HE2C3NPTBVWX3PGPLGRMJC2G3","json":"https://pith.science/pith/5HE2C3NPTBVWX3PGPLGRMJC2G3.json","graph_json":"https://pith.science/api/pith-number/5HE2C3NPTBVWX3PGPLGRMJC2G3/graph.json","events_json":"https://pith.science/api/pith-number/5HE2C3NPTBVWX3PGPLGRMJC2G3/events.json","paper":"https://pith.science/paper/5HE2C3NP"},"agent_actions":{"view_html":"https://pith.science/pith/5HE2C3NPTBVWX3PGPLGRMJC2G3","download_json":"https://pith.science/pith/5HE2C3NPTBVWX3PGPLGRMJC2G3.json","view_paper":"https://pith.science/paper/5HE2C3NP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1903.12225&json=true","fetch_graph":"https://pith.science/api/pith-number/5HE2C3NPTBVWX3PGPLGRMJC2G3/graph.json","fetch_events":"https://pith.science/api/pith-number/5HE2C3NPTBVWX3PGPLGRMJC2G3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5HE2C3NPTBVWX3PGPLGRMJC2G3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5HE2C3NPTBVWX3PGPLGRMJC2G3/action/storage_attestation","attest_author":"https://pith.science/pith/5HE2C3NPTBVWX3PGPLGRMJC2G3/action/author_attestation","sign_citation":"https://pith.science/pith/5HE2C3NPTBVWX3PGPLGRMJC2G3/action/citation_signature","submit_replication":"https://pith.science/pith/5HE2C3NPTBVWX3PGPLGRMJC2G3/action/replication_record"}},"created_at":"2026-05-17T23:49:54.550164+00:00","updated_at":"2026-05-17T23:49:54.550164+00:00"}