Signed Roman domination in digraphs

Abstract

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