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.