On a Generalization of Signed Total Dominating Functions of Graphs

Abstract

Let \(G = (V, E)\) be a simple graph. For any real valued function \(f: V \to \mathbb{R}\), the weight of \(f\) is defined as \(f(V) = \sum f(v)\), over all vertices \(v \in V\). For positive integer \(k\), a total \(k\)-subdominating function (TkSF) is a function \(f: V \to \{-1,1\}\) such that \(f(N(v)) \geq k\) for at least \(k\) vertices \(v\) of \(G\). The total \(k\)-subdomination number \(\gamma^t_{ks}(G)\) of a graph \(G\) equals the minimum weight of a TKSF on \(G\). In the special case where \(k = |V|\), \(\gamma^t_{ks}(G)\) is the signed total domination number \([5]\). We research total \(k\)-subdomination numbers of some graphs and obtain a few lower bounds of \(\gamma^t_{ks}(G)\).