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If these conditions hold and $G$ is a connected graph, then the set $\\mathfrak M_w$ of all such extensions is nonvoid and the shortest-path pseudometric $d_w$ is the greatest element of $\\mathfrak M_w$ with respect to the partial ordering $d_1 \\leqslant d_2$ if and only if $d_1(u,v) \\leqslant d_2(u,v)$ for all $u,v\\in V$. It is shown that every nonvoid poset $(\\mathfrak M_w,\\leqslant)$ contains the least element $\\"},"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":"1105.6167","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-05-31T06:25:40Z","cross_cats_sorted":[],"title_canon_sha256":"a0fa447bdf8bf35785903a06834bbcbb430da33809e149098f6774b77dba3480","abstract_canon_sha256":"b72301b676f864465016cd65b213c9c3dd01f1d02827fc505f47cb6d9f68e3a7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:21:00.020743Z","signature_b64":"NVczSVKqCcAoKmyGA2ufE4CeT61ZjZm7nuhr56igwsJ+h2P5Ajn54jGJu6Km+5g3/rN5lyuCajtRXqxUri3SAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"41bac1a28202b2dfde18c0a2c670cbeab5ae242679359ac72a45ac45e7bd2eb0","last_reissued_at":"2026-05-18T04:21:00.020052Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:21:00.020052Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Metrization of weighted graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Matti Vuorinen, Oleksiy Dovgoshey, Olli Martio","submitted_at":"2011-05-31T06:25:40Z","abstract_excerpt":"We find a set of necessary and sufficient conditions under which the weight $w:E\\to\\mathbb R^+$ on the graph $G=(V,E)$ can be extended to a pseudometric $d:V\\times V\\to\\mathbb R^+$. If these conditions hold and $G$ is a connected graph, then the set $\\mathfrak M_w$ of all such extensions is nonvoid and the shortest-path pseudometric $d_w$ is the greatest element of $\\mathfrak M_w$ with respect to the partial ordering $d_1 \\leqslant d_2$ if and only if $d_1(u,v) \\leqslant d_2(u,v)$ for all $u,v\\in V$. 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