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 ∈ N G (v) with f (u) > 0. The weight of a total Roman dominating function f on G is equal to f (V ) = v∈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(|V |) time.

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A function [Formula: see text] is a double Roman dominating function on a graph [Formula: see text] if for every vertex [Formula: see text] with [Formula: see text] either there is a vertex [Formula: see text] with [Formula: see text] or there are distinct vertices [Formula: see text] with [Formula: see text] and for every vertex [Formula: see text] with [Formula: see text] there is a vertex [Formula: see text] with [Formula: see text]. The weight of a double Roman dominating function [Formula: see text] on [Formula: see text] is the value [Formula: see text]. The minimum weight of a double Roman dominating function on [Formula: see text] is called the double Roman domination number of [Formula: see text]. In this paper, we give an algorithm to compute the double Roman domination number of a given proper interval graph [Formula: see text] in [Formula: see text] time.