Total Roman domination on the digraphs

Abstract

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

Keywords

• total roman dominating function
• total roman domination number
• digraph
• 05c20
• 05c69