Total Roman domination for proper interval graphs

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 is an element of V with f(v) = 0 there is a vertex u adjacent to v with f(u) = 2 and for every vertex v is an element of V with f(v) > 0 there exists a vertex u is an element of N-G(v) with f(u) > 0. The weight of a total Roman dominating function f on G is equal to f(V) = Sigma(v is an element of V) f(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(vertical bar V vertical bar) time.