Weak signed Roman k-domination in digraphs

Abstract

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