{"paper":{"title":"Critical Independent Sets of a Graph","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.DM","authors_text":"Eugen Mandrescu, Vadim E. Levit","submitted_at":"2014-07-28T09:36:29Z","abstract_excerpt":"Let $G$ be a simple graph with vertex set $V\\left( G\\right) $. A set $S\\subseteq V\\left( G\\right) $ is independent if no two vertices from $S$ are adjacent, and by $\\mathrm{Ind}(G)$ we mean the family of all independent sets of $G$.\n  The number $d\\left( X\\right) =$ $\\left\\vert X\\right\\vert -\\left\\vert N(X)\\right\\vert $ is the difference of $X\\subseteq V\\left( G\\right) $, and a set $A\\in\\mathrm{Ind}(G)$ is critical if $d(A)=\\max \\{d\\left( I\\right) :I\\in\\mathrm{Ind}(G)\\}$ (Zhang, 1990).\n  Let us recall the following definitions:\n  $\\mathrm{core}\\left( G\\right) $ = $\\bigcap$ {S : S is a maximum "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.7368","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"}