On signed arc total domination in digraphs

Abstract

Let $D=(V,A)$ be a finite simple digraph and $N(uv)=\{u^{\prime}v^{\prime}\neq uv \mid u=u^{\prime}\text{ or }v=v^{\prime}\}$ be the open neighbourhood of uv in $D$. A function $f: A\rightarrow \{-1, +1\}$ is said to be a signed arc total dominating function (SATDF) of $D$ if $\sum _{e^{\prime}\in N(uv)}f(e^{\prime})\geq 1$ holds for every arc $uv\in A$. The signed arc total domination number $\gamma^{\prime}_{st}(D)$ is defined as $\gamma^{\prime}_{st}(D)= \operatorname{min}\{\sum_{e\in A}f(e)\mid f \text{ is an SATDF of }D\}$. In this paper we initiate the study of the signed arc total domination in digraphs and present some lower bounds for this parameter.