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 ...

For any integer k ≥ 1, a signed (total) k-dominating function is a function f : V (G) → {−1, 1} satisfying w∈N [v] f(w) ≥ k ( P w∈N(v) f(w) ≥ k) for every v ∈ V (G), where N(v) = {u ∈ V (G)|uv ∈ E(G)} and N [v] = N(v)∪{v}. The minimum of the values ofv∈V (G) f(v), taken over all signed (total) k-dominating functions f, is called the signed (total) k-domination number and is denoted by γkS(G) (γ...

In his article published in 1999, Ian Stewart discussed a strategy of Emperor Constantine for defending the Roman Empire. Motivated by this article, Cockayne et al.(2004) introduced the notion of Roman domination in graphs. Let G = (V,E) be a graph. A Roman dominating function of G is a function f : V → {0, 1, 2} such that every vertex v for which f(v) = 0 has a neighbor u with f(u) = 2. The we...

The open neighborhood NG(e) of an edge e in a graph G is the set consisting of all edges having a common end-vertex with e and its closed neighborhood is NG[e] = NG(e) ∪ {e}. Let f be a function on E(G), the edge set of G, into the set {−1, 1}. If ∑x∈NG[e] f(x) ≥ 1 for at least a half of the edges e ∈ E(G), then f is called a signed edge majority dominating function of G. The minimum of the val...

For any integer $kgeq 1$ and any graph $G=(V,E)$ with minimum degree at least $k-1$‎, ‎we define a‎ ‎function $f:Vrightarrow {0,1,2}$ as a Roman $k$-tuple dominating‎ ‎function on $G$ if for any vertex $v$ with $f(v)=0$ there exist at least‎ ‎$k$ and for any vertex $v$ with $f(v)neq 0$ at least $k-1$ vertices in its neighborhood with $f(w)=2$‎. ‎The minimum weight of a Roman $k$-tuple dominatin...

Let G be a finite and simple graph with vertex set V (G), and let f: V (G)→ {−1, 1} be a two-valued function. If k > 1 is an integer and ∑ x∈N[v] f(x) > k for each v ∈ V (G), where N [v] is the closed neighborhood of v, then f is a signed k-dominating function on G. A set {f1, f2, . . . , fd} of signed kdominating functions on G with the property that ∑ d i=1 fi(x) 6 k for each x ∈ V (G), is ca...

Let D=(V(D),A(D)) be a finite, simple digraph and k positive integer. A function f:V(D)→{0,1,2,…,k+1} is called [k]-Roman dominating (for short, [k]-RDF) if f(AN−[v])≥|AN−(v)|+k for any vertex v∈V(D), where AN−(v)={u∈N−(v):f(u)≥1} AN−[v]=AN−(v)∪{v}. The weight of [k]-RDF f ω(f)=∑v∈V(D)f(v). minimum on D the domination number, denoted by γ[kR](D). For k=2 k=3, we call them double Roman number tr...

A double Roman dominating function on a graph $G$ with vertex set $V(G)$ is defined in cite{bhh} as a function$f:V(G)rightarrow{0,1,2,3}$ having the property that if $f(v)=0$, then the vertex $v$ must have at least twoneighbors assigned 2 under $f$ or one neighbor $w$ with $f(w)=3$, and if $f(v)=1$, then the vertex $v$ must haveat least one neighbor $u$ with $f(u)ge 2$. The weight of a double R...

A Roman dominating function on a graphG is a function f : V (G) → {0, 1, 2} satisfying the condition that every vertex u ∈ V (G) for which f(u) = 0 is adjacent to at least one vertex v ∈ V (G) for which f(v) = 2. The weight of a Roman dominating function is the value f(V (G)) = ∑ u∈V (G) f(u). The Roman domination number γR(G) of G is the minimum weight of a Roman dominating function on G. A Ro...

A Roman dominating function of a graph G is a labeling f : V (G) −→ {0, 1, 2} such that every vertex with label 0 has a neighbor with label 2. The Roman domination number γR(G) of G is the minimum of ∑ v∈V (G) f(v) over such functions. The Roman domination subdivision number sdγR(G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order t...