Signed Roman $k$-Domination in Graphs

Abstract

Let $k\ge 1$ be an integer, and let $G$ be a finite and simple graph with vertex set $V(G)$. A signed Roman $k$-dominating function (SRkDF) on a graph $G$ is a function $f:V(G)\rightarrow \{-1,1,2\}$ satisfying the conditions that (i) $\sum _{x\in N[v]}f(x)\ge k$ for each vertex $v\in V(G)$, where $N[v]$ is the closed neighborhood of $v$, and (ii) every vertex $u$ for which $f(u)=-1$ is adjacent to at least one vertex $v$ for which $f(v)=2$. The weight of an SRkDF $f$ is $w(f)=\sum _{v\in V(G)}f(v)$. The signed Roman $k$-domination number $\gamma _{sR}^k(G)$ of $G$ is the minimum weight of an SRkDF on $G$. In this paper we initiate the study of the signed Roman $k$-domination number of graphs, and we present different bounds on $\gamma _{sR}^k(G)$. In addition, we determine the signed Roman $k$-domination number of some classes of graphs. Some of our results are extensions of well-known properties of the signed Roman domination number $\gamma _{sR}(G)=\gamma _{sR}^1(G)$, introduced and investigated by Ahangar et al. (J Comb Optim doi:10.1007/s10878-012-9500-0, 2014).