Total Roman domination for proper interval graphs - Abolfazl Poureidi (GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB, 2020)
A function f:V → 0,1,2 is a total Roman dominating function (TRDF) on a graph G=(V,E) if for every vertex v ∈ V with f(v) = 0 there is a vertex u adjacent to v with f(u) = 2 and for every vertex v ∈ V with f(v) > 0 there exists a vertex u ∈ NG(v) with f(u) > 0. The weight of a total Roman dominating function f on G is equal to f(V)=Σv ∈ Vf(v). The minimum weight of a total Roman dominating function on G is called the total Roman domination number of G. In this paper, we give an algorithm to compute the total Roman domination number of a given proper interval graph G=(V,E) in O(|V|) time.