Minus $k$-subdomination in Graphs

Abstract

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