{"paper":{"title":"Embeddings into almost self-centered graphs of given radius","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Haiqiong Liu, Kexiang Xu, Kinkar Ch. Das, Sandi Klav\\v{z}ar","submitted_at":"2017-09-02T14:35:52Z","abstract_excerpt":"A graph is almost self-centered (ASC) if all but two of its vertices are central. An almost self-centered graph with radius $r$ is called an $r$-ASC graph. The $r$-ASC index $\\theta_r(G)$ of a graph $G$ is the minimum number of vertices needed to be added to $G$ such that an $r$-ASC graph is obtained that contains $G$ as an induced subgraph. It is proved that $\\theta_r(G)\\le 2r$ holds for any graph $G$ and any $r\\ge 2$ which improves the earlier known bound $\\theta_r(G)\\le 2r+1$. It is further proved that $\\theta_r(G)\\le 2r-1$ holds if $r\\geq 3$ and $G$ is of order at least $2$. The $3$-ASC in"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.00589","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"}