Domination in graphs has been an extensively researched branch of graph theory. Graph theory is one of the most flourishing branches of modern mathematics and computer applications. An introduction and an extensive overview on domination in graphs and related topics is surveyed and detailed in the two books by Haynes et al. [ 1, 2]. Recently dominating functions in domination theory have receiv...

Let G be a graph with vertex set V (G). A function f : V (G) → {−1, 1} is a signed dominating function of G if, for each vertex of G, the sum of the values of its neighbors and itself is positive. The signed domination number of a graph G, denoted γs(G), is the minimum value of ∑ v∈V (G) f(v) over all the signed dominating functions f of G. The signed reinforcement number of G, denoted Rs(G), i...

For any integer  ‎, ‎a minus  k-dominating function is a‎function  f‎ : ‎V (G)  {-1,0‎, ‎1} satisfying w) for every  vertex v, ‎where N(v) ={u V(G) | uv  E(G)}  and N[v] =N(v)cup {v}. ‎The minimum of ‎the values of  v)‎, ‎taken over all minus‎k-dominating functions f,‎ is called the minus k-domination‎number and is denoted by $gamma_k^-(G)$ ‎. ‎In this paper‎, ‎we ‎introduce the study of minu...

Let G be a graph with the vertex set V (G) and edge set E(G). A function f : E(G) → {−1,+1} is said to be a signed star dominating function ofG if ∑ e∈EG(v) f(e) ≥ 1, for every v ∈ V (G), where EG(v) = {uv ∈ E(G) |u ∈ V (G)}. The minimum of the values of ∑ e∈E(G) f(e), taken over all signed star dominating functions f on G is called the signed star domination number of G and is denoted by γss(G...

a function $f:v(g)rightarrow {-1,0,1}$ is a {em minusdominating function} if for every vertex $vin v(g)$, $sum_{uinn[v]}f(u)ge 1$. a minus dominating function $f$ of $g$ is calleda {em global minus dominating function} if $f$ is also a minusdominating function of the complement $overline{g}$ of $g$. the{em global minus domination number} $gamma_{g}^-(g)$ of $g$ isdefined as $gamma_{g}^-(g)=min{...

Let G be a graph with vertex set V ( ). A total Italian dominating function (TIDF) on is f : ) → {0, 1, 2} such that (i) every v = 0 adjacent to u 2 or two vertices w and z (ii) ≥ 1 1. The domination number γ tI the minimum weight of function. In this paper, we present Nordhaus–Gaddum type inequalities for number.

Given a graph G together with a capacity function c : V (G) → N, we call S ⊆ V (G) a capacitated dominating set if there exists a mapping f : (V (G) \ S) → S which maps every vertex in (V (G) \S) to one of its neighbors such that the total number of vertices mapped by f to any vertex v ∈ S does not exceed c(v). In the Planar Capacitated Dominating Set problem we are given a planar graph G, a ca...

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 ∑ x∈N[v] f (x) ≥ 1 for each v ∈ V (G), where N[v] is the closed neighborhood of v, then f is a signed dominating function on G. A set {f1, f2, . . . , fd} of signed dominating functions on Gwith the property that ∑d i=1 fi(x) ≤ 1 for each x ∈ V (G), is called a signed dominating fa...