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...

Abstract: Let G=(V,E) be a graph and let f:V(G)→{0,1,2} be a function‎. ‎A vertex v is protected with respect to f‎, ‎if f(v)>0 or f(v)=0 and v is adjacent to a vertex of positive weight‎. ‎The function f is a co-Roman dominating function‎, ‎abbreviated CRDF if‎: ‎(i) every vertex in V is protected‎, ‎and (ii) each u∈V with positive weight has a neighbor v∈V with f(v)=0 such that the func...

Recently, Zhang et al. [6] obtained some lower bounds of the signed domination number of a graph. In this paper, we obtain some new lower bounds of the signed domination number of a graph which are sharper than those of them.

A nonnegative signed dominating function (NNSDF) of a graph G is a function f from the vertex set V (G) to the set {−1, 1} such that ∑ u∈N [v] f(u) ≥ 0 for every vertex v ∈ V (G). The nonnegative signed domination number of G, denoted by γ s (G), is the minimum weight of a nonnegative signed dominating function on G. In this paper, we establish some sharp lower bounds on the nonnegative signed ...

Given a graph $G=(V,E)$ and a vertex $v in V$, by $N(v)$ we represent the open neighbourhood of $v$. Let $f:Vrightarrow {0,1,2}$ be a function on $G$. The weight of $f$ is $omega(f)=sum_{vin V}f(v)$ and let $V_i={vin V colon f(v)=i}$, for $i=0,1,2$. The function $f$ is said to bebegin{itemize}item a Roman ${2}$-dominating function, if for every vertex $vin V_0$, $sum_{uin N(v)}f(u)geq 2$. The R...

A Roman dominating function (RDF) on a graph $G$ is a function $f : V (G) to {0, 1, 2}$satisfying the condition that every vertex $u$ for which $f(u) = 0$ is adjacent to at least onevertex $v$ for which $f(v) = 2$. A Roman dominating function $f$ is called an outer-independentRoman dominating function (OIRDF) on $G$ if the set ${vin Vmid f(v)=0}$ is independent.The (outer-independent) Roman dom...

Let G = (V, E) be a graph. A function f : V → {−1,+1} defined on the vertices of G is a signed total dominating function if the sum of its function values over any open neighborhood is at least one. The signed total domination number of G, γ t (G), is the minimum weight of a signed total dominating function of G. In this paper, we study the signed total domination number of generalized Petersen...

Abstract. Let G = (V,E) be a simple graph. A function f : V → {−1, 1} is called an inverse signed total dominating function if the sum of its function values over any open neighborhood is at most zero. The inverse signed total domination number of G, denoted by γ0 st(G), equals to the maximum weight of an inverse signed total dominating function of G. In this paper, we establish upper bounds on...

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...

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...