A set D of vertices in a graph G is a distance-k dominating set if every vertex of G either is in D or is within distance k of at least one vertex in D. A distance-k dominating set of G of minimum cardinality is called a minimum distance-k dominating set of G. For any graph G and for a subset F of the edge set of G the set F is an edge dominating set of G if every edge of G either is in D or is...

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...

Vertex Cover (VC): A vertex cover in an undirected graph G = (V,E) is a subset of vertices V ′ ⊆ V such that every edge in G has at least one endpoint in V ′. The vertex cover problem (VC) is: given an undirected graph G and an integer k, does G have a vertex cover of size k? Dominating Set (DS): A dominating set in a graph G = (V,E) is a subset of vertices V ′ such that every vertex in the gra...

In a graph G, a vertex is said to dominate itself and all of its neighbors. For a fixed positive integer k , the k-tuple domination problem is to find a minimum sized vertex subset such that every vertex in the graph is dominated by at least k vertices in this set. The present paper studies the k-tuple domination problem in graphs from an algorithmic point of view. In particular, we give a line...

Vertex Cover (VC): A vertex cover in an undirected graph G = (V,E) is a subset of vertices V ′ ⊆ V such that every edge in G has at least one endpoint in V ′. The vertex cover problem (VC) is: given an undirected graph G and an integer k, does G have a vertex cover of size k? Dominating Set (DS): A dominating set in a graph G = (V,E) is a subset of vertices V ′ such that every vertex in the gra...

We consider a connected undirected graph G(n,m) with n nodes and m edges. A k-dominating set D in G is a set of nodes having the property that every node in G is at most k edges away from at least one node in D. Finding a k-dominating set of minimum size is NP-hard. We give a new synchronous distributed algorithm to find a k-dominating set in G of size no greater than ⌊n/(k + 1)⌋. Our algorithm...

Let D be a finite and simple digraph with vertex set V (D), and let f : V (D) → {−1, 1} be a two-valued function. If k ≥ 1 is an integer and ∑ x∈N−[v] f(x) ≥ k for each v ∈ V (D), where N−[v] consists of v and all vertices of D from which arcs go into v, then f is a signed k-dominating function on D. A set {f1, f2, . . . , fd} of distinct signed k-dominating functions of D with the property tha...

Given a graph G, a dominating set D is a set of vertices such that any vertex in G has at least one neighbor (or possibly itself) in D. A {k}-dominating multiset Dk is a multiset of vertices such that any vertex in G has at least k vertices from its closed neighborhood in Dk when counted with multiplicity. In this paper, we utilize the approach developed by Clark and Suen (2000) and properties ...

Top-k query is an efficient way to show the most important objects to user from massive amounts of data. After huge effort working on database performance, recently, an explain capability has attract more attention in recent years. In top-k query, since people may not specify his/her accurate preference, he may feel frustrated with the top-k results and propose a question such that “why my expe...

Let G = (V, E) be a graph. A k-connected restrained dominating set is a set S ⊆ V , where S is a restrained dominating set and G[S] has at most k components. The k-connected restrained domination number of G, denoted by γ k r (G), is the smallest cardi-nality of a k-connected restrained dominating set of G. In this talk, I will give some exact values and sharp bounds for γ k r (G). Then the nec...