Let k ≥ 1 be an integer. A signed Roman k-dominating function on a digraph D is a function f : V (D) −→ {−1, 1, 2} such that ∑x∈N−[v] f(x) ≥ k for every v ∈ V (D), where N−[v] consists of v and all in-neighbors of v, and every vertex u ∈ V (D) for which f(u) = −1 has an in-neighbor w for which f(w) = 2. A set {f1, f2, . . . , fd} of distinct signed Roman k-dominating functions on D with the pro...

Let D be a finite and simple digraph with the vertex set V (D), and let f : V (D) → {−1, 1} be a two-valued function. If∑ x∈N[v] f(x) ≥ 1 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 dominating function on D. The sum f(V (D)) is called the weight w(f) of f . The minimum of weights w(f), taken over all signed dominating function...

Let G = BHn be a n - dimensional balanced hypercube. As topology of interconnection network, hypercubes are widely used in many areas. The signed k subdomination number graphs is an important parameter the domination theory. In this paper, according to properties hypercubes, (|G| –1) when 2 determined by classified discussion and exhaustived method.

Let γ ′ s (G) be the signed edge domination number of G. In 2006, Xu conjectured that: for any 2-connected graph G of order n(n ≥ 2), γ ′ s (G) ≥ 1. In this article we show that this conjecture is not true. More precisely, we show that for any positive integer m, there exists an m-connected graph G such that γ ′ s (G) ≤ − m 6 |V (G)|. Also for every two natural numbers m and n, we determine γ ′...