A Roman dominating function (RDF) on a graphG = (V,E) is a function f : V −→ {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. The weight of an RDF is the value f(V (G)) = ∑ u∈V (G) f(u). An RDF f in a graph G is independent if no two vertices assigned positive values are adjacent. The Roman domination number γR(G)...

The complementary prism GG of a graph G is formed from the disjoint union of G and its complement G by adding the edges of a perfect matching between the corresponding vertices of G and G. A Roman dominating function on a graph G = (V,E) is a labeling f : V (G) → {0, 1, 2} such that every vertex with label 0 is adjacent to a vertex with label 2. The Roman domination number γR(G) ofG is the mini...

LetD = (V,A) be a finite and simple digraph. A Roman dominating function (RDF) on a digraph D is a labeling f : V (D) → {0, 1, 2} such that every vertex with label 0 has a in-neighbor with label 2. The weight of an RDF f is the value ω(f) = ∑ v∈V f(v). The Roman domination number of a digraph D, denoted by γR(D), equals the minimum weight of an RDF on D. In this paper we present some sharp boun...

A Roman dominating function (RDF) on a graph $G = (V, E)$ is a labeling $f : V rightarrow {0, 1, 2}$ suchthat every vertex with label $0$ has a neighbor with label $2$. The weight of $f$ is the value $f(V) = Sigma_{vin V} f(v)$The Roman domination number, $gamma_R(G)$, of $G$ is theminimum weight of an RDF on $G$.An RDF of minimum weight is called a $gamma_R$-function.A graph G is said to be $g...

Let G be a graph with no isolated vertex and f : V ( ) → {0, 1, 2} function. i = { x ∈ } for every . We say that is total Roman dominating function on if in 0 adjacent to at least one 2 the subgraph induced by 1 ∪ has vertex. The weight of ω ∑ v minimum among all functions domination number , denoted γ t R It known general problem computing NP-hard. In this paper, we show H nontrivial graph, th...

A Roman dominating function of a graph G is a function f : V → {0, 1, 2} such that every vertex with 0 has a neighbor with 2. The minimum of f (V (G)) = ∑ v∈V f (v) over all such functions is called the Roman domination number γR(G). A 2-rainbow dominating function of a graphG is a function g that assigns to each vertex a set of colors chosen from the set {1, 2}, for each vertex v ∈ V (G) such ...