Using a dominating set as a coordinator in wireless networks has been proposed in many papers as an energy conservation technique. Since the nodes in a dominating set have the extra burden of coordination, energy resources in such nodes will drain out more quickly than in other nodes. To maximize the lifetime of nodes in the network, it has been proposed that the role of coordinators be rotated...

For a positive integer k, a k-rainbow dominating function of a graph G is a function f from the vertex set V (G) to the set of all subsets of the set {1, 2, . . . , k} such that for any vertex v ∈ V (G) with f(v) = ∅ the condition ⋃ u∈N(v) f(u) = {1, 2, . . . , k} is fulfilled, where N(v) is the neighborhood of v. The 1-rainbow domination is the same as the ordinary domination. A set {f1, f2, ....

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...

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...

A partition of V (G), all of whose classes are dominating sets in G, is called a domatic partition of G. The maximum number of classes of a domatic partition of G is called the domatic number of G. The concept of a domatic number was introduced in [1]. More interesting results on domatically full graphs, domatically critical, domatically cocritical graphs and other domatic numbers can be found ...

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...

‎a set $s$ of vertices of a graph $g=(v,e)$ without isolated vertex‎ ‎is a {em total dominating set} if every vertex of $v(g)$ is‎ ‎adjacent to some vertex in $s$‎. ‎the {em total domatic number} of‎ ‎a graph $g$ is the maximum number of total dominating sets into‎ ‎which the vertex set of $g$ can be partitioned‎. ‎we show that the‎ ‎total domatic number of a random $r$-regular graph is almost‎...