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The proper subset $W=\\{w_1,\\ldots,w_{|W|}\\}$ is a {\\em completeness-resolving set} of $G$ if\n  $$\n  \\Psi_W: V(G)\\setminus W \\longrightarrow [m(W)]^{|W|},\\qquad u\\longmapsto (d(w_1,u),\\ldots,d(w_{|W|},u))\n  $$ is a bijection, where\n  $$\n  [m(W)]^{|W|}=\\{(a_{(1)},\\ldots,a_{(|W|)})\\mid 1\\leq a_{(i)}\\leq m(W)\\text{ for each }i=1,"},"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":"2101.02838","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2021-01-08T03:57:55Z","cross_cats_sorted":[],"title_canon_sha256":"ef57a0639d6cf29b7097a7aefdb54d654e95dcfd3eab8bcae09eaef36c22a63a","abstract_canon_sha256":"66f5d225294f228b438ea01236f77fb52a18f248e78ac04093c0e10787a457b3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:05:30.935781Z","signature_b64":"kXeKskF1oV+GuiIK9FdDH0eBs/ljFPsL6N2n9RrwcT2qV28ekqvnm7Ulks7GjrrhugKIXx5YU+jQzRjPZcE4Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4c433fd38d60481af30bfeae22933a76af7075ebf9debb1575f64c17212d2cd6","last_reissued_at":"2026-07-05T02:05:30.935439Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:05:30.935439Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Completeness-resolvable graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Huiling Xu, Min Feng, Xuanlong Ma","submitted_at":"2021-01-08T03:57:55Z","abstract_excerpt":"Given a connected graph $G=(V(G), E(G))$, the length of a shortest path from a vertex $u$ to a vertex $v$ is denoted by $d(u,v)$. For a proper subset $W$ of $V(G)$, let $m(W)$ be the maximum value of $d(u,v)$ as $u$ ranging over $W$ and $v$ ranging over $V(G)\\setminus W$. The proper subset $W=\\{w_1,\\ldots,w_{|W|}\\}$ is a {\\em completeness-resolving set} of $G$ if\n  $$\n  \\Psi_W: V(G)\\setminus W \\longrightarrow [m(W)]^{|W|},\\qquad u\\longmapsto (d(w_1,u),\\ldots,d(w_{|W|},u))\n  $$ is a bijection, where\n  $$\n  [m(W)]^{|W|}=\\{(a_{(1)},\\ldots,a_{(|W|)})\\mid 1\\leq a_{(i)}\\leq m(W)\\text{ for each }i=1,"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2101.02838","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2101.02838/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":"2101.02838","created_at":"2026-07-05T02:05:30.935493+00:00"},{"alias_kind":"arxiv_version","alias_value":"2101.02838v1","created_at":"2026-07-05T02:05:30.935493+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2101.02838","created_at":"2026-07-05T02:05:30.935493+00:00"},{"alias_kind":"pith_short_12","alias_value":"JRBT7U4NMBEB","created_at":"2026-07-05T02:05:30.935493+00:00"},{"alias_kind":"pith_short_16","alias_value":"JRBT7U4NMBEBV4YL","created_at":"2026-07-05T02:05:30.935493+00:00"},{"alias_kind":"pith_short_8","alias_value":"JRBT7U4N","created_at":"2026-07-05T02:05:30.935493+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/JRBT7U4NMBEBV4YL72XCFEZ2O2","json":"https://pith.science/pith/JRBT7U4NMBEBV4YL72XCFEZ2O2.json","graph_json":"https://pith.science/api/pith-number/JRBT7U4NMBEBV4YL72XCFEZ2O2/graph.json","events_json":"https://pith.science/api/pith-number/JRBT7U4NMBEBV4YL72XCFEZ2O2/events.json","paper":"https://pith.science/paper/JRBT7U4N"},"agent_actions":{"view_html":"https://pith.science/pith/JRBT7U4NMBEBV4YL72XCFEZ2O2","download_json":"https://pith.science/pith/JRBT7U4NMBEBV4YL72XCFEZ2O2.json","view_paper":"https://pith.science/paper/JRBT7U4N","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2101.02838&json=true","fetch_graph":"https://pith.science/api/pith-number/JRBT7U4NMBEBV4YL72XCFEZ2O2/graph.json","fetch_events":"https://pith.science/api/pith-number/JRBT7U4NMBEBV4YL72XCFEZ2O2/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/JRBT7U4NMBEBV4YL72XCFEZ2O2/action/timestamp_anchor","attest_storage":"https://pith.science/pith/JRBT7U4NMBEBV4YL72XCFEZ2O2/action/storage_attestation","attest_author":"https://pith.science/pith/JRBT7U4NMBEBV4YL72XCFEZ2O2/action/author_attestation","sign_citation":"https://pith.science/pith/JRBT7U4NMBEBV4YL72XCFEZ2O2/action/citation_signature","submit_replication":"https://pith.science/pith/JRBT7U4NMBEBV4YL72XCFEZ2O2/action/replication_record"}},"created_at":"2026-07-05T02:05:30.935493+00:00","updated_at":"2026-07-05T02:05:30.935493+00:00"}