{"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"}