Total Roman domination for proper interval graphs

Abstract

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 )=Σ 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.