Wang, Haichao; Tong, Jie; Volkmann, Lutz A note on signed total 2-independence in graphs. (English) Zbl 1238.05205 Util. Math. 85, 213-223 (2011). For any vertex \(v\) of a graph \(G\), \(N(v)\) denotes the set of vertices adjacent to \(v\). A signed total 2-independence function \(f:V(G)\to\{-1,+1\}\) on a graph \(G\) without isolated vertices satisfies \(\sum\limits_{u\in N(v)}f(u)\geq1\) for all \(v\in V(G)\). The weight \(w(f)\) of such a function is \(\sum_{v\in V(G)}f(v)\); the maximum weight over all signed total 2-independence functions on \(G\) is the signed total 2-independence number \(\alpha^2_{st}(G)\); \(\alpha^2_{st}\) is called the negative decision number in [C. Wang, “The negative decision number in graphs”, Australas. J. Comb. 41, 263–272 (2008; Zbl 1154.05050)]. From the authors’ introduction: “In this paper we continue the study on \(\alpha^2_{st}(G)\), and present some sharp bounds…for general graphs and \(K_{r+1}\)-free graphs. Some known bounds on \(\alpha^2_{st}(G)\) are improved or extended.” Reviewer: William G. Brown (MontrĂ©al) Cited in 1 Document MSC: 05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.) 05C22 Signed and weighted graphs 05C35 Extremal problems in graph theory Keywords:bounds; signed total 2-independence number; \(K_{r+1}\)-free graphs Citations:Zbl 1154.05050 PDFBibTeX XMLCite \textit{H. Wang} et al., Util. Math. 85, 213--223 (2011; Zbl 1238.05205)