{"paper":{"title":"Roman domination and Mycieleki's structure in graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Adel P. Kazemi","submitted_at":"2011-05-17T07:11:38Z","abstract_excerpt":"For a graph $G=(V,E)$, a function $f:V\\rightarrow \\{0,1,2\\}$ is called Roman dominating function (RDF) if for any vertex $v$ with $f(v)=0$, there is at least one vertex $w$ in its neighborhood with $f(w)=2$. The weight of an RDF $f$ of $G$ is the value $f(V)=\\sum_{v\\in V}f(v)$. The minimum weight of an RDF of $G$ is its Roman domination number and denoted by $\\gamma_ R(G)$. In this paper, we first show that $\\gamma_{R}(G)+1\\leq \\gamma_{R}(\\mu (G))\\leq \\gamma_{R}(G)+2$, where $\\mu (G)$ is the Mycielekian graph of $G$, and then characterize the graphs achieving equality in these bounds. Then for"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1105.3290","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"}