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The boundary $\\partial(G)$ of $G$ is the set of all of its boundary vertices.\n  The boundary distance matrix $\\hat{D}_G$ of a graph $G=([n],E)$ is the square matrix of order $\\kappa$, being $\\kappa$ the order of $\\partial(G)$, such that for every $i,j\\in \\partial(G)$, $[\\hat{D}_G]_{ij}=d_G(i,j)$.\n  Given a square matrix $\\hat{B}$ of order $\\kappa$, we prove under which conditions $\\hat{B}$ is the distance matrix $\\hat{D}_T$ of t"},"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":"2404.04039","kind":"arxiv","version":8},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.CO","submitted_at":"2024-04-05T11:49:28Z","cross_cats_sorted":[],"title_canon_sha256":"0ec145a503b6017bb9f47b8ee0d56b16a903d376ec9b39843a5c35ce6e2a01e4","abstract_canon_sha256":"1d1dd0ed9058952546c075ef430e24f2e8446db9c21395ae3712412152a223dd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:54:26.003669Z","signature_b64":"lmRFlHgQP+aj/9fzOLy3PX54TGnQnWlzaBAKg1wLvwzghYjr1qjwzDbvL+boCOn77j76qKCqQkT+PrB3Wz8cCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"36c7f9e9a3db41429eb05f1049dc42a0d1b5bc71a1e02372139ae7036cf1dd69","last_reissued_at":"2026-07-05T09:54:26.003237Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:54:26.003237Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Reconstructing a graph from the distance matrix of its boundary","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ignacio M. Pelayo, Jos\\'e C\\'aceres","submitted_at":"2024-04-05T11:49:28Z","abstract_excerpt":"A vertex $v$ of a connected graph $G$ is said to be a boundary vertex of $G$ if for some other vertex $u$ of $G$, no neighbor of $v$ is further away from $u$ than $v$. The boundary $\\partial(G)$ of $G$ is the set of all of its boundary vertices.\n  The boundary distance matrix $\\hat{D}_G$ of a graph $G=([n],E)$ is the square matrix of order $\\kappa$, being $\\kappa$ the order of $\\partial(G)$, such that for every $i,j\\in \\partial(G)$, $[\\hat{D}_G]_{ij}=d_G(i,j)$.\n  Given a square matrix $\\hat{B}$ of order $\\kappa$, we prove under which conditions $\\hat{B}$ is the distance matrix $\\hat{D}_T$ of t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2404.04039","kind":"arxiv","version":8},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2404.04039/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2404.04039","created_at":"2026-07-05T09:54:26.003288+00:00"},{"alias_kind":"arxiv_version","alias_value":"2404.04039v8","created_at":"2026-07-05T09:54:26.003288+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2404.04039","created_at":"2026-07-05T09:54:26.003288+00:00"},{"alias_kind":"pith_short_12","alias_value":"G3D7T2ND3NAU","created_at":"2026-07-05T09:54:26.003288+00:00"},{"alias_kind":"pith_short_16","alias_value":"G3D7T2ND3NAUFHVQ","created_at":"2026-07-05T09:54:26.003288+00:00"},{"alias_kind":"pith_short_8","alias_value":"G3D7T2ND","created_at":"2026-07-05T09:54:26.003288+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/G3D7T2ND3NAUFHVQL4IETXCCUD","json":"https://pith.science/pith/G3D7T2ND3NAUFHVQL4IETXCCUD.json","graph_json":"https://pith.science/api/pith-number/G3D7T2ND3NAUFHVQL4IETXCCUD/graph.json","events_json":"https://pith.science/api/pith-number/G3D7T2ND3NAUFHVQL4IETXCCUD/events.json","paper":"https://pith.science/paper/G3D7T2ND"},"agent_actions":{"view_html":"https://pith.science/pith/G3D7T2ND3NAUFHVQL4IETXCCUD","download_json":"https://pith.science/pith/G3D7T2ND3NAUFHVQL4IETXCCUD.json","view_paper":"https://pith.science/paper/G3D7T2ND","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2404.04039&json=true","fetch_graph":"https://pith.science/api/pith-number/G3D7T2ND3NAUFHVQL4IETXCCUD/graph.json","fetch_events":"https://pith.science/api/pith-number/G3D7T2ND3NAUFHVQL4IETXCCUD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/G3D7T2ND3NAUFHVQL4IETXCCUD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/G3D7T2ND3NAUFHVQL4IETXCCUD/action/storage_attestation","attest_author":"https://pith.science/pith/G3D7T2ND3NAUFHVQL4IETXCCUD/action/author_attestation","sign_citation":"https://pith.science/pith/G3D7T2ND3NAUFHVQL4IETXCCUD/action/citation_signature","submit_replication":"https://pith.science/pith/G3D7T2ND3NAUFHVQL4IETXCCUD/action/replication_record"}},"created_at":"2026-07-05T09:54:26.003288+00:00","updated_at":"2026-07-05T09:54:26.003288+00:00"}