on hop roman domination in trees

Let g=(v,e) be a graph. a subset s⊂v is a hop dominating set if every vertex outside s is at distance two from a vertex of s. a hop dominating set s which induces a connected subgraph is called a connected hop dominating set of g. the connected hop domination number of g, γch(g), is the minimum cardinality of a connected hop dominating set of g. a hop roman dominating function (hrdf) of a graph g is a function f:v(g)⟶{0,1,2} having the property that for every vertex v∈v with f(v)=0 there is a vertex u with f(u)=2 and d(u,v)=2. the weight of an hrdf f is the sum f(v)=∑v∈vf(v). the minimum weight of an hrdf on g is called the hop roman domination number of g and is denoted by γhr(g). we give an algorithm that decides whether γhr(t)=2γch(t) for a given tree t.