a {em 2-rainbow dominating function} (2rdf) of a graph $g$ is a function $f$ from the vertex set $v(g)$ to the set of all subsets of the set ${1,2}$ such that for any vertex $vin v(g)$ with $f(v)=emptyset$ the condition $bigcup_{uin n(v)}f(u)={1,2}$ is fulfilled, where $n(v)$ is the open neighborhood of $v$. the {em weight} of a 2rdf $f$ is the value $omega(f)=sum_{vin v}|f (v)|$. the {em $2$-r...

Top-k dominating queries combine the advantages of top-k queries and skyline queries, and eliminate their disadvantages. They return k objects with the highest domination score, which is defined as the number of dominated objects. As a top-k query, the user can bound the number of returned results through the parameter k, and like a skyline query a user-selected scoring function is not required...

An Italian dominating function (IDF) of a graph G is f : V(G) → {0, 1, 2} satisfying the condition that for every v ∈ V with f(v) = 0, Σu∈N(v)f(u) ≥ 2. The weight an IDF on sum f(V) Σv∈Vf(v) and domination number, γI (G), minimum IDF. perfect (PID) G, if vertex 0 total assigned by to neighbours exactly 2, i.e., all u are except one which 2 or two vertices w f(w) 1. PID-function Σu∈V(G) f(u). nu...

Let G = (V,E) be a simple and undirected graph. For some integer k > 1, a set D ⊆ V is said to be a k-dominating set in G if every vertex v of G outside D has at least k neighbors in D. Furthermore, for some real number α with 0 < α 6 1, a set D ⊆ V is called an α-dominating set in G if every vertex v of G outside D has at least α×dv neighbors in D, where dv is the degree of v in G. The cardina...

A set D ⊆ V of vertices is said to be a (connected) distance k-dominating set of G if the distance between each vertex u ∈ V − D and D is at most k (and D induces a connected graph in G). The minimum cardinality of a (connected) distance k-dominating set in G is the (connected) distance k-domination number of G, denoted by γk(G) (γ c k (G), respectively). The set D is defined to be a total k-do...

Let S be a dataset of n 2-dimensional points. The top-k dominating query aims to report the k points that dominate the most points in S . A point p dominates a point q iff all coordinates of p are smaller than or equal to those of q and at least one of them is strictly smaller. The top-k dominating query combines the dominance concept of maxima queries with the ranking function of top-k queries...

A subset S of a vertex set of a graphG is a total (k, r)-dominating set if every vertex u ∈ V (G) is within distance k of at least r vertices in S. The minimum cardinality among all total (k, r)-dominating sets ofG is called the total (k, r)domination number of G, denoted by γ (k,r)(G). We previously gave an upper bound on γ t (2,r)(G(n, p)) in random graphs with non-fixed p ∈ (0, 1). In this p...

A k-dominating set is a set D k V such that every vertex i 2 V nD k has at least k i neighbours in D k. The k-domination number k (G) of G is the cardinality of a smallest k-dominating set of G. For k 1 = ::: = k n = 1, k-domination corresponds to the usual concept of domination. Our approach yields an improvement of an upper bound for the domination number found then the conception of k-domina...