— In this paper, we study the dominating-free set which is defined as follows: k points called servers and n points called clients in the plane are given. For a point p in the plane is said to be dominated by a client c if for every server s, the distance between s and p is greater than the distance between s and c. The dominating-free set is the set of points in the plane which aren't dominate...

A set D ⊆ V of a graph G = (V,E) is called a restrained dominating set of G if every vertex not in D is adjacent to a vertex in D and to a vertex in V \D. The MINIMUM RESTRAINED DOMINATION problem is to find a restrained dominating set of minimum cardinality. Given a graph G, and a positive integer k, the RESTRAINED DOMINATION DECISION problem is to decide whether G has a restrained dominating ...

A subset X of edges in a graph G is called an edge dominating set of G if every edge not in X is adjacent to some edge in X. The edge domination number γ′(G) of G is the minimum cardinality taken over all edge dominating sets of G. Let m,n and k be positive integers with n − 1 ≤ m ≤ (n 2 ) , G(m,n) be the set of all non-isomorphic connected graphs of order n and size m, and G(m,n; k) = {G ∈ G(m...

Let k be a positive integer and G = (V,E) a connected graph of order n. A set D ⊆ V is called a distance k-dominating set of G if each x ∈ V (G)−D is within distance k from some vertex of D. The k-domination number of G, denoted by γk(G), is the minimum cardinality over all distance k-dominating sets. Determining γk(G) has a significant impact on an efficient design of routing protocols in netw...

Let G = (V (G), E(G)) be a simple graph, and let k be a positive integer. A subset D of V (G) is a k-dominating set if every vertex of V (G) − D is dominated at least k times by D. The k-domination number γk(G) is the minimum cardinality of a k-dominating set of G. In [5] Volkmann showed that for every nontrivial tree T, γ2(T ) ≥ γ1(T ) + 1 and characterized extremal trees attaining this bound....

For any graph, there is a largest integer k such that given any partition of the vertex set with at most k elements in each class of the partition, there is transversal of the partition that is a dominating set in the graph. Some basic results about this parameter, the partition domination number, are obtained. In particular, it is shown that its value is 2 for the two-dimensional infinite grid...

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 on D with the property that ...

The dominating set problem is a core NP-hard problem in combinatorial optimization and graph theory, and has many important applications. Baker [JACM 41,1994] introduces a k-outer planar graph decomposition-based framework for designing polynomial time approximation scheme (PTAS) for a class of NP-hard problems in planar graphs. It is mentioned that the framework can be applied to obtain an O(2...

We explore a reconfiguration version of the dominating set problem, where a dominating set in a graph G is a set S of vertices such that each vertex is either in S or has a neighbour in S. In a reconfiguration problem, the goal is to determine whether there exists a sequence of feasible solutions connecting given feasible solutions s and t such that each pair of consecutive solutions is adjacen...