The signed total Roman domatic number of a digraph

Abstract

Let $D$ be a finite and simple digraph with vertex set $V(D)$. A signed total Roman dominating function on the digraph $D$ is a function $f:V(D)⟶-1,1,2$ $\sum _{u\in N-(v)}f(u)\ge 1$ for every $v\in V(D)$, where $N^{-}(v)$ consists of all inner neighbors of $v$ for dominating function on $D$ with the property that $\sum _d^{i=1}{f}_i(v)\le 1$ for each $v\in V(D)$ is called a signed total roman dominating family (of functions) on $D$. The maximum number of functions in a signed total roman dominating family on $D$is the signed total Roman domatic number of $D$. denoted by $d_{stR}(D)$. In addition, we determine the signed total Roman domatic number of some digraphs. Some of our results are extensions of well-known properties of the signed total Roman domatic of graphs.