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Abstract
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:V(G)→{0,1,2} satisfying the condition that every vertex u for which f(u)=0 is adjacent to at least one vertex v for which f(v)=2. A Roman dominating function f is called an independent Roman dominating function if the set of all vertices with positive weights is an independent set. The weight of an independent Roman dominating function f is the value f(V(G))=∑u∈V(G)f(u). The independent Roman domination number of G, denoted by iR(G), is the minimum weight of an independent Roman dominating function on G. A subset S of V is a 2-independent set of G if every vertex of S has at most one neighbor in S. The maximum cardinality of a 2-independent set of G is the 2-independence number β2(G). These two parameters are incomparable in general, however, we show that for any tree T, β2(T)≥iR(T) and we characterize all trees attaining the equality.
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