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{...

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...

‎for any integer $kge 1$‎, ‎a minus $k$-dominating function is a‎ ‎function $f‎ : ‎v (g)rightarrow {-1,0‎, ‎1}$ satisfying $sum_{win‎‎n[v]} f(w)ge k$ for every $vin v(g)$‎, ‎where $n(v) ={u in‎‎v(g)mid uvin e(g)}$ and $n[v] =n(v)cup {v}$‎. ‎the minimum of‎‎the values of $sum_{vin v(g)}f(v)$‎, ‎taken over all minus‎‎$k$-dominating functions $f$‎, ‎is called the minus $k$-domination‎‎number and i...

A function f de1ned on the vertices of a graph G = (V; E); f :V → {−1; 0; 1} is a minus dominating function if the sum of its values over any closed neighborhood is at least one. The weight of a minus dominating function is f(V ) = ∑ v∈V f(v). The minus domination number of a graph G, denoted by −(G), equals the minimum weight of a minus dominating function of G. In this paper, a sharp lower bo...

A function f : V (G) → {+1,−1} defined on the vertices of a graph G is a signed dominating function if for any vertex v the sum of function values over its closed neighborhood is at least 1. The signed domination number γs(G) of G is the minimum weight of a signed dominating function on G. By simply changing “{+1,−1}” in the above definition to “{+1, 0,−1}”, we can define the minus dominating f...

3 A function f : V (G) → {+1,−1} defined on the vertices of a graph G is a signed domi4 nating function if for any vertex v the sum of function values over its closed neighborhood 5 is at least one. The signed domination number γs(G) of G is the minimum weight of a 6 signed dominating function on G. By simply changing “{+1,−1}” in the above definition 7 to “{+1, 0,−1}”, we can define the minus ...

A function f : V → {−1, 0, 1} is a minus-domination function of a graph G = (V,E) if the values over the vertices in each closed neighborhood sum to a positive number. The weight of f is the sum of f(x) over all vertices x ∈ V. The minus-domination number γ(G) is the minimum weight over all minus-domination functions. The size of a minus domination is the number of vertices that are assigned 1....

A function f : V (G) → {−1, 0, 1} is a minus dominating function if for every vertex v ∈ V (G), ∑ u∈N [v] f(u) ≥ 1. A minus dominating function f of G is called a global minus dominating function if f is also a minus dominating function of the complement G of G. The global minus domination number γ− g (G) of G is defined as γ − g (G) = min{ ∑ v∈V (G) f(v) | f is a global minus dominating functi...