Given a graph G together with a capacity function c : V (G) → N, we call S ⊆ V (G) a capacitated dominating set if there exists a mapping f : (V (G) \ S) → S which maps every vertex in (V (G) \S) to one of its neighbors such that the total number of vertices mapped by f to any vertex v ∈ S does not exceed c(v). In the Planar Capacitated Dominating Set problem we are given a planar graph G, a ca...

For a positive integer k, a total {k}-dominating function of a digraph D is a function f from the vertex set V (D) to the set {0, 1, 2, . . . , k} such that for any vertex v ∈ V (D), the condition ∑ u∈N(v) f(u) ≥ k is fulfilled, where N(v) consists of all vertices of D from which arcs go into v. A set {f1, f2, . . . , fd} of total {k}-dominating functions of D with the property that ∑ d i=1 fi(...

In this paper, the concept of a strong n-Connected Total Perfect k-connected total perfect k-dominating set and weak n-connected in fuzzy graphs is introduced. current work, triple-connected dominating modified to an nctpkD(G) number γnctpD(G). New definitions are compared with old ones. Strong obtained. The results those sets discussed spanning graphs, arcs, sets, generalization complete, conn...

Let k~l be an integer, and let G = (V, E) be a graph. The closed kneighborhood N k[V] of a vertex v E V is the set of vertices within distance k from v. A 3-valued function f defined on V of the form f : V --+ { -1,0, I} is a three-valued k-neighborhood dominating function if the sum of its function values over any closed k-neighborhood is at least 1. The weight of a threevalued k-neighborhood ...

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...

The in-neighborhood, I(v), of a vertex v in a digraph D=(V, A) is v together with the set of all vertices sending an arc to v, i.e., vertices u such that (u, v) # A. A subset of V is called dominating if it meets I(v) for every v # V. (To avoid confusion, it must be noted that some authors require in the definition meeting every out-neighborhood.) A set of vertices is called independent if no t...

Let be a simple graph with vertex set and edge set . Let have at least vertices of degree at least , where and b are positive integers. A function is said to be a signed -edge cover of G if G ( ) V G ( ) e E v ( ) E G G : ( f E k b k ) { 1,1} G   ( , ) b k ( ) f e b    for at least vertices of , where . The value k v G ( ) = {uv E( ( ) E v G u N v   ) | } ( ) min ( ) G e E f e   , taki...

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, ....

In this paper, we study the Dominating Set problem in random graphs. In a random graph, each pair of vertices are joined by an edge with a probability of p, where p is a positive constant less than 1. We show that, given a random graph in n vertices, a minimum dominating set in the graph can be computed in expected 2 2 2 n) time. For the parameterized dominating set problem, we show that it can...