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The Steiner distance $\\mathrm{sd}_G(A)$ of $A \\subseteq V(G)$ within $G$ is the minimum length of a connected subgraph of $G$ containing $A$, where the length of a subgraph is the sum of the lengths of its edges.\n  It is clear that every subgraph $H \\subseteq G$, with the induced length-function $\\ell|_{E(H)}$, satisfies $\\mathrm{sd}_H(A) \\geq \\mathrm{sd}_G(A)$ for every $A \\subseteq V(H)$. We call $H \\subseteq G$ $k$-geodesic in $G$ if equality is attained for every $A \\subseteq V(H)$ with $"},"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":"1703.09969","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-03-29T11:00:17Z","cross_cats_sorted":[],"title_canon_sha256":"fcfe6380b63d251ce9b647e104a7ed0f8511fb4719c32329ec601cd89e50e9e8","abstract_canon_sha256":"2db43edd547450ef14de91137ae774ff35f259c9453e84a695f385a1fad0c0b1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:47:38.915445Z","signature_b64":"aqdeh9GGolEKM/qCepfPVWx5EMANlEt3vB1PJE6yu96exiTDITt91AfASAXtU8KBckQ5vq1+DfSf/Y9Igoe8Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4090fe4442643307d75300183092df88d8c701a546b8d9bcba932e7c5ba57b19","last_reissued_at":"2026-05-18T00:47:38.914767Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:47:38.914767Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Steiner trees and higher geodecity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Daniel Wei{\\ss}auer","submitted_at":"2017-03-29T11:00:17Z","abstract_excerpt":"Let $G$ be a connected graph and $\\ell : E(G) \\to \\mathbb{R}^+$ a length-function on the edges of $G$. 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We call $H \\subseteq G$ $k$-geodesic in $G$ if equality is attained for every $A \\subseteq V(H)$ with $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.09969","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":"1703.09969","created_at":"2026-05-18T00:47:38.914852+00:00"},{"alias_kind":"arxiv_version","alias_value":"1703.09969v1","created_at":"2026-05-18T00:47:38.914852+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1703.09969","created_at":"2026-05-18T00:47:38.914852+00:00"},{"alias_kind":"pith_short_12","alias_value":"ICIP4RCCMQZQ","created_at":"2026-05-18T12:31:21.493067+00:00"},{"alias_kind":"pith_short_16","alias_value":"ICIP4RCCMQZQPV2T","created_at":"2026-05-18T12:31:21.493067+00:00"},{"alias_kind":"pith_short_8","alias_value":"ICIP4RCC","created_at":"2026-05-18T12:31:21.493067+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/ICIP4RCCMQZQPV2TAAMDBEW7RD","json":"https://pith.science/pith/ICIP4RCCMQZQPV2TAAMDBEW7RD.json","graph_json":"https://pith.science/api/pith-number/ICIP4RCCMQZQPV2TAAMDBEW7RD/graph.json","events_json":"https://pith.science/api/pith-number/ICIP4RCCMQZQPV2TAAMDBEW7RD/events.json","paper":"https://pith.science/paper/ICIP4RCC"},"agent_actions":{"view_html":"https://pith.science/pith/ICIP4RCCMQZQPV2TAAMDBEW7RD","download_json":"https://pith.science/pith/ICIP4RCCMQZQPV2TAAMDBEW7RD.json","view_paper":"https://pith.science/paper/ICIP4RCC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1703.09969&json=true","fetch_graph":"https://pith.science/api/pith-number/ICIP4RCCMQZQPV2TAAMDBEW7RD/graph.json","fetch_events":"https://pith.science/api/pith-number/ICIP4RCCMQZQPV2TAAMDBEW7RD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ICIP4RCCMQZQPV2TAAMDBEW7RD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ICIP4RCCMQZQPV2TAAMDBEW7RD/action/storage_attestation","attest_author":"https://pith.science/pith/ICIP4RCCMQZQPV2TAAMDBEW7RD/action/author_attestation","sign_citation":"https://pith.science/pith/ICIP4RCCMQZQPV2TAAMDBEW7RD/action/citation_signature","submit_replication":"https://pith.science/pith/ICIP4RCCMQZQPV2TAAMDBEW7RD/action/replication_record"}},"created_at":"2026-05-18T00:47:38.914852+00:00","updated_at":"2026-05-18T00:47:38.914852+00:00"}