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.