Signed Roman k-domination in Digraphs

Abstract

Let $k\ge 1$ be an integer, and let D be a finite and simple digraph with vertex set V(D). A signed Roman k-dominating function (SRkDF) on a digraph D is a function $f:V(D)\rightarrow \{-1,1,2\}$ satisfying the conditions that (1) $\sum _{x\in N^-[v]}f(x)\ge k$ for each $v\in V(D)$, where $N^-[v]$ consists of v and all vertices of D from which arcs go into v, and (2) every vertex u for which $f(u)=-1$ has an inner neighbor v for which $f(v)=2$. The weight of an SRkDF f is $w(f)=\sum _{v\in V(D)}f(v)$. The signed Roman k-domination number $\gamma _{sR}^k(D)$ of D is the minimum weight of an SRkDF on D. In this paper we initiate the study of the signed Roman k-domination number of digraphs, and we present different bounds on $\gamma _{sR}^k(D)$. In addition, we determine the signed Roman k-domination number of some classes of digraphs. Some of our results are extensions of well-known properties of the signed Roman domination number $\gamma _{sR}(D)=\gamma _{sR}^1(D)$ and the signed Roman k-domination number $\gamma _{sR}^k(G)$ of graphs G.