On the restrained Roman domination in graphs
نویسندگان
چکیده
A Roman dominating function (RDF) on a graph G is a function f : V (G) → {0, 1, 2} satisfying the condition that every vertex v for which f(v) = 0, is adjacent to at least one vertex u for which f(u) = 2. The weight of a Roman dominating function f is the value f(V (G)) = ∑ v∈V (G) f(v). The Roman domination number of G, denoted by γR(G), is the minimum weight of an RDF on G. For a given graph, a Roman dominating function f = (V0, V1, V2) is a restrained Roman dominating function (rRDF) if every vertex of V0 has a neighbor in V0. The restrained Roman domination number of G, denoted by γrR(G), is the minimum weight of an rRDF on G. We first show that the restrained Roman domination problem is NP-complete. Then we give various bounds and characterizations. Finally we study restrained Roman domination in random graphs.
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