Total Roman domination on the digraphs

Abstract

Let $D=\left(V,A\right)$ be a simple digraph with vertex set $V$, arc set $A$, and no isolated vertex. A total Roman dominating function (TRDF) of $D$ is a function $h:V\to \left\{0,1,2\right\}$, which satisfies that each vertex $x\in V$ with $h\left(x\right)=0$ has an in-neighbour $y\in V$ with $h(y)=2$, and that the subdigraph of $D$ induced by the set $\left\{x\in V:h\left(x\right)\ge 1\right\}$ has no isolated vertex. The weight of a TRDF $h$ is $\omega \left(h\right)={\sum }_{x\in V}h\left(x\right)$. The total Roman domination number ${\gamma }_{tR}\left(D\right)$ of $D$ is the minimum weight of all TRDFs of $D$. The concept of TRDF on a graph $G$ was introduced by Liu and Chang [Roman domination on strongly chordal graphs, J. Comb. Optim. 26 (2013), no. 3, 608–619]. In 2019, Hao et al. [Total Roman domination in digraphs, Quaest. Math. 44 (2021), no. 3, 351–368] generalized the concept to digraph and characterized the digraphs of order $n\ge 2$ with ${\gamma }_{tR}\left(D\right)=2$ and the digraphs of order $n\ge 3$ with ${\gamma }_{tR}\left(D\right)=3$. In this article, we completely characterize the digraphs of order $n\ge k$ with ${\gamma }_{tR}\left(D\right)=k$ for all integers $k\ge 4$, which generalizes the results mentioned above.