let $d$ be a finite and simple digraph with vertex set $v(d)$‎.‎a signed total roman $k$-dominating function (str$k$df) on‎‎$d$ is a function $f:v(d)rightarrow{-1‎, ‎1‎, ‎2}$ satisfying the conditions‎‎that (i) $sum_{xin n^{-}(v)}f(x)ge k$ for each‎‎$vin v(d)$‎, ‎where $n^{-}(v)$ consists of all vertices of $d$ from‎‎which arcs go into $v$‎, ‎and (ii) every vertex $u$ for which‎‎$f(u)=-1$ has a...

Let $kge 1$ be an integer, and let $G$ be a finite and simple graph with vertex set $V(G)$.A weak signed Roman $k$-dominating function (WSRkDF) on a graph $G$ is a function$f:V(G)rightarrow{-1,1,2}$ satisfying the conditions that $sum_{xin N[v]}f(x)ge k$ for eachvertex $vin V(G)$, where $N[v]$ is the closed neighborhood of $v$. The weight of a WSRkDF $f$ is$w(f)=sum_{vin V(G)}f(v)$. The weak si...

Let D be a finite and simple digraph with vertex set V (D), and let f : V (D) → {−1, 1} be a two-valued function. If k ≥ 1 is an integer and x∈N−[v] f(x) ≥ k for each v ∈ V (D), where N−[v] consists of v and all vertices of D from which arcs go into v, then f is a signed k-dominating function on D. A set {f1, f2, . . . , fd} of distinct signed k-dominating functions on D with the property that ...

Let G be a simple graph without isolated vertices with vertex set V (G) and edge set E(G) and let k be a positive integer. A function f : E(G) → {−1, 1} is said to be a signed star k-dominating function on G if ∑ e∈E(v) f(e) ≥ k for every vertex v of G, where E(v) = {uv ∈ E(G) | u ∈ N(v)}. A set {f1, f2, . . . , fd} of signed star k-dominating functions on G with the property that ∑d i=1 fi(e) ...

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 neighborhood of v, then f is a signed total k-dominating function on G. A set {f1, f2, . . . , fd} of signed total k-dominating functions on G with the property that ∑d i=1 fi(x) ≤ k for each x ∈ V (G), ...

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...

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...

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...

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 ...

Let k ≥ 1 be an integer, and let G be a finite and simple graph with vertex set V (G). A signed total Italian k-dominating function (STIkDF) on a graph G is a functionf : V (G) → {−1, 1, 2} satisfying the conditions that $sum_{xin N(v)}f(x)ge k$ for each vertex v ∈ V (G), where N(v) is the neighborhood of $v$, and each vertex u with f(u)=-1 is adjacent to a vertex v with f(v)=2 or to two vertic...