For an integer n ≥ 2, let I ⊂ {0, 1, 2, · · · , n}. A Smarandachely Roman sdominating function for an integer s, 2 ≤ s ≤ n on a graph G = (V,E) is a function f : V → {0, 1, 2, · · · , n} satisfying the condition that |f(u)− f(v)| ≥ s for each edge uv ∈ E with f(u) or f(v) ∈ I . Similarly, a Smarandachely Roman edge s-dominating function for an integer s, 2 ≤ s ≤ n on a graph G = (V,E) is a func...

A Roman dominating function (RDF) on a graph G = (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 f is the value f(V (G)) = ∑ u∈V (G) f(u). A function f : V (G) → {0, 1, 2} with the ordered partition (V0, V1, V2) of V (G), where Vi = {v ∈ V (G) | f(v) = i} for i = 0...

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)...

Let γ(G) denote the domination number of a graph G. A Roman domination 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 Roman domination number γR(G) is the minimum of f(V (G)) = Σv∈V f(v) over all such functions. Let G H denote the Cartesian product of graphs G and H. We prove that γ(G)γ(H) ≤ γR(G H) for all simple graphs G and H, w...

let $d$ be a finite and simple digraph with vertex set $v(d)$‎.‎a signed total roman $k$-dominating function (str$k$df) on‎‎$d$ is a function $f:v(d)rightarrow{-1‎, ‎1‎, ‎2}$ satisfying the conditions‎‎that (i) $sum_{xin n^{-}(v)}f(x)ge k$ for each‎‎$vin v(d)$‎, ‎where $n^{-}(v)$ consists of all vertices of $d$ from‎‎which arcs go into $v$‎, ‎and (ii) every vertex $u$ for which‎‎$f(u)=-1$ has a...

A {em Roman dominating function} on a graph $G$ is a function $f:V(G)rightarrow {0,1,2}$ 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 {em total Roman dominating function} is a Roman dominating function with the additional property that the subgraph of $G$ induced by the set of all vertices of positive weight has n...

‎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 ...