{"paper":{"title":"On the Signed (Total) $k$-Domination Number of a Graph","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.DM","authors_text":"Hongyu Liang","submitted_at":"2012-04-21T16:58:00Z","abstract_excerpt":"Let $k$ be a positive integer and $G=(V,E)$ be a graph of minimum degree at least $k-1$. A function $f:V\\rightarrow \\{-1,1\\}$ is called a \\emph{signed $k$-dominating function} of $G$ if $\\sum_{u\\in N_G[v]}f(u)\\geq k$ for all $v\\in V$. The \\emph{signed $k$-domination number} of $G$ is the minimum value of $\\sum_{v\\in V}f(v)$ taken over all signed $k$-dominating functions of $G$. The \\emph{signed total $k$-dominating function} and \\emph{signed total $k$-domination number} of $G$ can be similarly defined by changing the closed neighborhood $N_G[v]$ to the open neighborhood $N_G(v)$ in the definit"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.4827","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"}