Outer independent Roman domination number of trees
A Roman dominating function (RDF) on a graph G=(V,E) is a function f : V → {0, 1, 2} such that every vertex u for which f(u)=0 is adjacent to at least one vertex v for which f(v)=2. An RDF f is calledan outer independent Roman dominating function (OIRDF) if the set ofvertices assigned a 0 under f is an independent set. The weight of anOIRDF is the sum of its function values over all vertices, and the outerindependent Roman domination number ΥoiR (G) is the minimum weightof an OIRDF on $G$. In this paper, we show that if T is a tree of order n ≥ 3 with s(T) support vertices, then $gamma _{oiR}(T)leq min {frac{5n}{6},frac{3n+s(T)}{4}}.$ Moreover, we characterize the tressattaining each bound.
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