Total Roman domination on the digraphs

Let D = (V, A) 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? {0, 1, 2}, which satisfies that each vertex x ? Vwith h(x) = 0 has an in-neighbour y ? Vwith h(y) = 2, and that the subdigraph of D induced by the set {x ? V : h(x) = 1} has no isolated vertex. The weight of a TRDF h is ?(h) = S(x?V)h(x). The total Roman domination number ?(tR)(D) 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 = 2 with ?(tR)(D) = 2 and the digraphs of order n = 3 with ?(tR)(D) = 3. In this article, we completely characterize the digraphs of order n = k with ?(tR)(D) = k for all integers k =4, which generalizes the results mentioned above.