The minus $k$-domination numbers in graphs

Abstract

For any integer $k\ge 1$‎, ‎a minus $k$-dominating function is a function $f‎ : ‎V \rightarrow \{-1,0‎, ‎1\}$ satisfying $\sum_{w\in N[v]} f(w)\ge k$ for every $v\in V(G)$‎, ‎where $N(v) =\{u \in V(G)\mid uv\in E(G)\}$ and $N[v] =N(v)\cup \{v\}$‎. ‎The minimum of the values of $\sum_{v\in V(G)}f(v)$‎, ‎taken over all minus‎ ‎$k$-dominating functions $f$‎, ‎is called the minus $k$-domination number and is denoted by $\gamma^-_{k}(G)$‎. ‎In this paper‎, ‎we introduce the study of minus $k$-domination in graphs and present several sharp lower bounds on the minus $k$-domination number for general graphs‎.

Keywords

• Minus $k$-dominating function‎
• ‎Minus $k$-domination number.