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...

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 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...

Efficient routing among mobile hosts is an important function in ad hoc networks. Routing based on a connected dominating set is a promising approach, where the search space for a route is reduced to the hosts in the set. A set is dominating if all the hosts are either in the set or neighbors of hosts in the set. The efficiency of dominating-set-based routing mainly depends on the overhead intr...

In standard kernelization algorithms, the usual goal is to reduce, in polynomial time, an instance (I, k) of a parameterized problem to an equivalent instance (I ′, k′) of size bounded by a function in k. One of the central problems in this area, whose investigation has led to the development of many kernelization techniques, is the Dominating Set problem. Given a graph G and k ∈ N, Dominating ...

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(...

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...