A map f : V → {0, 1, 2} is a Roman dominating function on a graph G = (V, E) if for every vertex v ∈ V with f (v) = 0, there exists a vertex u, adjacent to v, such that f (u) = 2. The weight of a Roman dominating function is given by f (V ) = u∈V f (u). The minimum weight of a Roman dominating function on G is called the Roman domination number of G. In this article we study the Roman domination number of Generalized Sierpiński graphs S(G, t). More precisely, we obtain a general upper bound on the Roman domination number of S(G, t) and we discuss the tightness of this bound. In particular, we focus on the cases in which the base graph G is a path, a cycle, a complete graph or a graph having exactly one universal vertex.
In this paper we obtain closed formulae for several parameters of generalized Sierpiński graphs S(G, t) in terms of parameters of the base graph G. In particular, we focus on the chromatic, vertex cover, clique and domination numbers.
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