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An assignment of capacities to the edges of $G$ and demands to the edges of $H$ is said to satisfy the \\emph{cut condition} if for any cut in the graph, the total demand crossing the cut is no more than the total capacity crossing it. The pair $(G,H)$ is called \\emph{cut-sufficient} if for any assignment of capacities and demands that satisfy the cut condition, there is a multiflow routing the demands defined on $H$ within the network with capacities defined on $G$. We prove a previous conjecture,"},"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":"1203.4041","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2012-03-19T06:25:33Z","cross_cats_sorted":["cs.DS"],"title_canon_sha256":"896deaae046956201f45ec9b8b4ff9ea9caa34cbb3e6ffd6898cf467608832b7","abstract_canon_sha256":"93f61d1d8c77e132951d00854c9e6292f3ce881d6df66572f834d5b591d1096c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:59:48.433117Z","signature_b64":"3kHFilJs08+4yx3ULMtqUUAmcq1pyaehqQ9U+cUo2ea1NyR3vfP/yN+JqatJEvMvua4dnDz97x7kP/aXZYOnDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"325087459fdd5f777ba1a210eb0c6f46045402c8a46d09d5bcd4981570059f49","last_reissued_at":"2026-05-18T03:59:48.432742Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:59:48.432742Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"When the Cut Condition is Enough: A Complete Characterization for Multiflow Problems in Series-Parallel Networks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS"],"primary_cat":"cs.DM","authors_text":"Amit Chakrabarti, Christophe Weibel, Lisa Fleischer","submitted_at":"2012-03-19T06:25:33Z","abstract_excerpt":"Let $G=(V,E)$ be a supply graph and $H=(V,F)$ a demand graph defined on the same set of vertices. 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