A subset D of the vertex set V (G) of a graph G is called point-set dominating, if for each subset S ⊆ V (G) − D there exists a vertex v ∈ D such that the subgraph of G induced by S ∪ {v} is connected. The maximum number of classes of a partition of V (G), all of whose classes are point-set dominating sets, is the point-set domatic number dp(G) of G. Its basic properties are studied in the paper.

A subset D of V (G) is called an equitable dominating set of a graph G if for every v ∈ (V − D), there exists a vertex u ∈ D such that uv ∈ E(G) and |deg(u) − deg(v)| 6 1. The minimum cardinality of such a dominating set is denoted by γe(G) and is called equitable domination number of G. In this paper we introduce the equitable edge domination and equitable edge domatic number in a graph, exact...

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 graph. A subset D of V is called common neighbourhood dominating set (CN-dominating set) if for every v ∈ V −D there exists a vertex u ∈ D such that uv ∈ E(G) and |Γ(u, v)| > 1, where |Γ(u, v)| is the number of common neighbourhood between the vertices u and v. The minimum cardinality of such CN-dominating set denoted by γcn(G) and is called common neighbourhood domination n...

A Roman dominating function (RDF) on a digraph $D$ is a function $f: V(D)rightarrow {0,1,2}$ satisfying the condition that every vertex $v$ with $f(v)=0$ has an in-neighbor $u$ with $f(u)=2$. The weight of an RDF $f$ is the value $sum_{vin V(D)}f(v)$. The Roman domination number of a digraph $D$ is the minimum weight of an RDF on $D$. A set ${f_1,f_2,dots,f_d}$ of Roman dominating functions on ...

Let k ≥ j ≥ 1 be two integers, and letG be a simple graph such that j(δ(G)+1) ≥ k, where δ(G) is the minimum degree of G. A (j, k)-dominating function of a graph G is a function f from the vertex set V (G) to the set {0, 1, 2, . . . , j} such that for any vertex v ∈ V (G), the condition ∑ u∈N[v] f(u) ≥ k is fulfilled, where N [v] is the closed neighborhood of v. A set {f1, f2, . . . , fd} of (j...

Network lifetime is a critical issue in wireless sensor networks. In the coverage problem, sensors can be partitioned into many subsets to prolong network lifetime. These subsets are activated successively and each of them completely covers an interest region. Many centralized algorithms have been proposed to solve this problem. A very few distributed versions have also been presented but none ...

Let G be a (p, q)-graph with edge domination number γ′ and edge domatic number d′. In this paper we characterize connected graphs for which γ′ = p/2 and graphs for which γ′ + d′ = q + 1. We also characterize trees and unicyclic graphs for which γ′ = bp/2c and γ′ = q −∆′, where ∆′ denotes the maximum degree of an edge in G.