Total Roman domination on the digraphs
Xinhong Zhang, Xin Song, Ruijuan Li · Open Mathematics · 2023
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 ) =