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We determine the complexity of finding a $p$-partition $(V_1, \\dots, V_p)$ of the vertex set of a given digraph such that the maximum out-degree of each of the digraphs induced by $V_i$, ($1\\leq i\\leq p$) is at least $k$ smaller than the maximum out-degree of $D$. We show that this problem is polynomial-time solvable when $p\\geq 2k$ and ${\\cal NP}$-complete otherwise. The result for $k=1$ and $p=2$ answers a question posed in \\cite{bangTCS636}. We also determine, for all fixed non-negative integers $k_1,k_2,p$, the complexity of deciding whether a given digraph of m"},"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":"1707.09349","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2017-07-28T17:45:33Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"57f3eb86f8a6cf36282030d27b2ecd78f4a67323218e36b88ecd16b0edb9b885","abstract_canon_sha256":"908a9b07cd0e2be046e4f5dcd2ca68183d659e382d046b1c5d577d03123402b9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:39:16.431084Z","signature_b64":"WW3yOfBl5eR9sSKGhmwPyo5glgKz3dxOYUOrTJUmJFIEpzpNkxYi39auf6fQA5gwFrPLbCf/oEZq9vZJANUiCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9e753df335cc805169cbc7046e55c6fdb6c96a3cd7fd3502aa29e5b270713f6e","last_reissued_at":"2026-05-18T00:39:16.430412Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:39:16.430412Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Out-degree reducing partitions of digraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.DM","authors_text":"Anders Yeo, Fr\\'ed\\'eric Havet, Joergen Bang-Jensen, St\\'ephane Bessy","submitted_at":"2017-07-28T17:45:33Z","abstract_excerpt":"Let $k$ be a fixed integer. We determine the complexity of finding a $p$-partition $(V_1, \\dots, V_p)$ of the vertex set of a given digraph such that the maximum out-degree of each of the digraphs induced by $V_i$, ($1\\leq i\\leq p$) is at least $k$ smaller than the maximum out-degree of $D$. We show that this problem is polynomial-time solvable when $p\\geq 2k$ and ${\\cal NP}$-complete otherwise. The result for $k=1$ and $p=2$ answers a question posed in \\cite{bangTCS636}. 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