Let $D$ be a finite and simple digraph with vertex set $V(D)$. A weak signed Roman dominating function (WSRDF) on is $f:V(D)\rightarrow\{-1,1,2\}$ satisfying the condition that $\sum_{x\in N^-[v]}f(x)\ge 1$ for each $v\in V(D)$, where $N^-[v]$ consists of $v$ allvertices from which arcs go into $v$. The weight WSRDF $f$ $\sum_{v\in V(D)}f(v)$. domination number $\gamma_{wsR}(D)$ minimum $D$. In...

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...

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 R be a commutative ring (with 1) and let Z(R) be its set of zero-divisors. The zero-divisor graph Γ(R) has vertex set Z∗(R) = Z(R) \ {0} and for distinct x, y ∈ Z∗(R), the vertices x and y are adjacent if and only if xy = 0. In this paper, we consider the domination number and signed domination number on zero-divisor graph Γ(R) of commutative ring R such that for every 0 6= x ∈ Z∗(R), x 6= ...

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...

For a nonempty graph G = (V, E), a signed edge-domination of G is a function f : E(G) → {1,−1} such that ∑e′∈NG [e] f (e′) ≥ 1 for each e ∈ E(G). The signed edge-domatic number of G is the largest integer d for which there is a set { f1, f2, . . . , fd} of signed edge-dominations of G such that ∑d i=1 fi (e) ≤ 1 for every e ∈ E(G). This paper gives an original study on this concept and determin...

Let G = (V, E) be a simple graph on vertex set V and define a function f : V → {−1, 1}. The function f is a signed dominating function if for every vertex x ∈ V , the closed neighborhood of x contains more vertices with function value 1 than with −1. The signed domination number of G, γs(G), is the minimum weight of a signed dominating function on G. Let G denote the complement of G. In this pa...