{"paper":{"title":"Some Results on [1, k]-sets of Lexicographic Products of Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DM","authors_text":"M.R. Hooshmandasl, P. Sharifani","submitted_at":"2017-08-01T09:30:40Z","abstract_excerpt":"A subset $S \\subseteq V$ in a graph $G = (V,E)$ is called a $[1, k]$-set, if for every vertex $v \\in V \\setminus S$, $1 \\leq | N_G(v) \\cap S | \\leq k$. The $[1,k]$-domination number of $G$, denoted by $\\gamma_{[1, k]}(G)$ is the size of the smallest $[1,k]$-sets of $G$. A set $S'\\subseteq V(G)$ is called a total $[1,k]$-set, if for every vertex $v \\in V$, $1 \\leq | N_G(v) \\cap S | \\leq k$. If a graph $G$ has at least one total $[1, k]$-set then the cardinality of the smallest such set is denoted by $\\gamma_{t[1, k]}(G)$. We consider $[1, k]$-sets that are also independent. Note that not every "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.00219","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"}