Total Roman domination on the digraphs

Abstract

Abstract Let D = ( V , A ) D=\left(V,A) be a simple digraph with vertex set V V , arc set A A , and no isolated vertex. A total Roman dominating function (TRDF) of D D is a function h : V → { 0 , 1 , 2 } h:V\to \left\{0,1,2\right\} , which satisfies that each vertex x ∈ V x\in V with h ( x ) = 0 h\left(x)=0 has an in-neighbour y ∈ V y\in V with h ( y ) = 2 h(y)=2 , and that the subdigraph of D D induced by the set { x ∈ V : h ( x ) ≥ 1 } \left\{x\in V:h\left(x)\ge 1\right\} has no isolated vertex. The weight of a TRDF h h is ω ( h ) = ∑ x ∈ V h ( x ) \omega \left(h)={\sum }_{x\in V}h\left(x) . The total Roman domination number γ t R ( D ) {\gamma }_{tR}\left(D) of D D is the minimum weight of all TRDFs of D D . The concept of TRDF on a graph G 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 n\ge 2 with γ t R ( D ) = 2 {\gamma }_{tR}\left(D)=2 and the digraphs of order n ≥ 3 n\ge 3 with γ t R ( D ) = 3 {\gamma }_{tR}\left(D)=3 . In this article, we completely characterize the digraphs of order n ≥ k n\ge k with γ t R ( D ) = k {\gamma }_{tR}\left(D)=k for all integers k ≥ 4 k\ge 4 , which generalizes the results mentioned above.