Let k be a positive integer, and let G be a simple graph with vertex set V (G). A k-distance Roman dominating function on G is a labeling f : V (G) → {0, 1, 2} such that every vertex with label 0 has a vertex with label 2 within distance k from each other. A set {f1, f2, . . . , fd} of k-distance Roman dominating functions on G with the property that ∑d i=1 fi(v) ≤ 2 for each v ∈ V (G), is call...

A Roman dominating function on a graph G is a function f : V (G) → {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 a Roman dominating function is the value f(V (G)) =

A Roman dominating function (RDF) on a digraph $D$ is a function $f: V(D)rightarrow {0,1,2}$ satisfying the condition that every vertex $v$ with $f(v)=0$ has an in-neighbor $u$ with $f(u)=2$. The weight of an RDF $f$ is the value $sum_{vin V(D)}f(v)$. The Roman domination number of a digraph $D$ is the minimum weight of an RDF on $D$. A set ${f_1,f_2,dots,f_d}$ of Roman dominating functions on ...

A Roman dominating function of a graph G = (V, E) is a function f : V → {0, 1, 2} such that every vertex x with f (x) = 0 is adjacent to at least one vertex y with f (y) = 2. The weight of a Roman dominating function is defined to be f (V ) = ∑ x∈V f (x), and the minimum weight of a Roman dominating function on a graph G is called the Roman domination number of G. In this paper we first answer ...

For a graph G = (V,E), a Roman dominating function on G is a function f : V (G) → {0, 1, 2} such that every vertex u for which f(u) = 0 is adjacent to at least one vertex v with f(v) = 2. The weight of a Roman dominating function is the value f(V (G)) = ∑ u∈V (G) f(u). The minimum weight of a Roman dominating function on a graph G is called the Roman domination number of G, denoted by γR (G). T...

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. The Roman reinforc...

‎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 ...

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...

‎in this paper‎, ‎we define the common minimal dominating signed‎ ‎graph of a given signed graph and offer a structural‎ ‎characterization of common minimal dominating signed graphs‎. ‎in‎ ‎the sequel‎, ‎we also obtained switching equivalence‎ ‎characterizations‎: ‎$overline{s} sim cmd(s)$ and $cmd(s) sim‎ ‎n(s)$‎, ‎where $overline{s}$‎, ‎$cmd(s)$ and $n(s)$ are complementary‎ ‎signed gra...

A Roman dominating function on a graph G = (V,E) is a function f : V −→ {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 is the value w(f) = ∑ v∈V f(v). The Roman domination number of a graph G, denoted by γR(G), equals the minimum weight of a Roman dominating function on ...