Minus \(k\)-subdomination in Graphs

Let \(G = (V, E)\) be a graph and \(k \in \mathbb{Z}^+\) such that \(1 \leq k \leq |V|\). A \(k\)-subdominating function (KSF) to \(\{-1, 0, 1\}\) is a function \(f: V \to \{-1, 0, 1\}\) such that the closed neighborhood sum \(f(N[v]) \geq 1\) for at least \(k\) vertices of \(G\). The weight of a KSF \(f\) is \(f(V) = \sum_{v \in V} f(v)\). The \(k\)-subdomination number to \(\{-1, 0, 1\}\) of a graph \(G\), denoted by \(\gamma^{-101}_{k_s}(G)\), equals the minimum weight of a KSF of \(G\). In this paper, we characterize minimal KSF’s, calculate \(\gamma^{-101}_{k_s}(G)\) for an arbitrary path \(P_n\), and determine the least order of a connected graph \(G\) for which \(\gamma^{-101}_{k_s}(G)=-m\) for an arbitrary positive integer \(m\).