<![CDATA[A subset S of vertices of G is a hop dominating set if every vertex outside S is at distance two from a vertex of S. The hop domination number of G, denoted by γh(G), is the minimum cardinality of a hop dominating set of G. A hop Roman dominating function (HRDF) of 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 a HRDF f is the sum f(V) = ∑v∈Vf(v). The minimum weight of HRDF on G is called the hop Roman domination number of G, and is denoted by γhR(G). In this paper we characterize all graphs G with γhR(G) = γh(G), all trees T for which γhR(T) = γh(T) + 1 as well as all trees T for which γhR(T) = γh(T) + 2.]]>
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