Efficient algorithms for independent Roman domination on some classes of graphs

Let G=(V,E) be a given graph of order n. A function f:V→{0,1,2} is an independent Roman dominating function (IRDF) on G if for every vertex v∈V with f(v)=0 there is a vertex u adjacent to v with f(u)=2 and {v∈V:f(v)>0} is an independent set. The weight of an IRDF f on G is the value f(V)=∑v∈Vf(v). The minimum weight of an IRDF among all IRDFs on G is called the independent Roman domination number of~G. In this paper, we give algorithms for computing the independent Roman domination number of G in O(|V|) time when G=(V,E) is a tree, unicyclic graph or proper interval graph.