Let G = (V,E) be a simple graph. A subset Dof V (G) is a (k, r)dominating set if for every vertexv ∈ V −D, there exists at least k vertices in D which are at a distance utmost r from v in [1]. The minimum cardinality of a (k, r)-dominating set of G is called the (k, r)-domination number of G and is denoted by γ(k,r)(G). In this paper, minimal (k, r)dominating sets are characterized. It is prove...

We consider finite graphs G with vertex set V (G). A subset D ⊆ V (G) is a dominating set of the graph G, if every vertex v ∈ V (G) − D is adjacent to at least one vertex in D. The domination number γ(G) is the minimum cardinality among the dominating sets of G. In this note, we characterize the trees T with an even number of vertices such that γ(T ) = |V (T )| − 2

The last years there is an increasing interest for query processing techniques that take into consideration the dominance relationship between objects to select the most promising ones, based on user preferences. Skyline and top-k dominating queries are examples of such techniques. A skyline query computes the objects that are not dominated, whereas a top-k dominating query returns the k object...

For a positive integer k, a k-subdominating function of a graph G=(V; E) is a function f :V →{−1; 1} such that ∑u∈NG [v] f(u)¿ 1 for at least k vertices v of G. The ksubdomination number of G, denoted by ks(G), is the minimum of ∑ v∈V f(v) taken over all k-subdominating functions f of G. In this article, we prove a conjecture for k-subdomination on trees proposed by Cockayne and Mynhardt. We al...

A vertex v in a graph G dominates itself as well as its neighbors. A set S of vertices in G is (1) a dominating set if every vertex of G is dominated by some vertex of S, (2) an open dominating set if every vertex of G is dominated by a vertex of S distinct from itself, and (3) a double dominating set if every vertex of G is dominated by at least two distinct vertices of S. The minimum cardinal...

Let k be a positive integer. A subset S of V (G) in a graph G is a k-tuple total dominating set of G if every vertex of G has at least k neighbors in S. The k-tuple total domination number γ×k,t(G) of G is the minimum cardinality of a k-tuple total dominating set of G. In this paper for a given graph G with minimum degree at least k, we find some sharp lower and upper bounds on the k-tuple tota...

The domination number γ(G) of a graph G is the minimum cardinality of a subset D of V (G) with the property that each vertex of V (G) − D is adjacent to at least one vertex of D. For a graph G with n vertices we define ǫ(G) to be the number of leaves in G minus the number of stems in G, and we define the leaf density ζ(G) to equal ǫ(G)/n. We prove that for any graph G with no isolated vertex, γ...

The domination number γ(H) of a hypergraph H = (V (H), E(H)) is the minimum size of a subset D ⊂ V (H) of the vertices such that for every v ∈ V (H) \D there exist a vertex d ∈ D and an edge H ∈ E(H) with v, d ∈ H. We address the problem of finding the minimum number n(k, γ) of vertices that a k-uniform hypergraph H can have if γ(H) ≥ γ and H does not contain isolated vertices. We prove that n(...

Let K n denote the Cartesian product Kn Kn Kn, where Kn is the complete graph on n vertices. We show that the domination number of K n

given a graph $g$, the total dominator coloring problem seeks aproper coloring of $g$ with the additional property that everyvertex in the graph is adjacent to all vertices of a color class. weseek to minimize the number of color classes. we initiate to studythis problem on several classes of graphs, as well as findinggeneral bounds and characterizations. we also compare the totaldominator chro...