Given a graph G, a dominating set D is a set of vertices such that any vertex not in D has at least one neighbor 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) to prove a ‘‘Vizing-like’’ inequ...

A 2-rainbow dominating function ( ) of a graph  is a function  from the vertex set  to the set of all subsets of the set  such that for any vertex  with  the condition  is fulfilled, where  is the open neighborhood of . A maximal 2-rainbow dominating function on a graph  is a 2-rainbow dominating function  such that the set is not a dominating set of . The weight of a maximal    is the value . ...

The k-dominating graph Dk(G) of a graph G is defined on the vertex set consisting of dominating sets of G with cardinality at most k, two such sets being adjacent if they differ by either adding or deleting a single vertex. In this paper, after presenting several basic properties of k-dominating graphs, it is proved that if G is a graph with no isolates, of order n ≥ 2, and with G ∼= Dk(G), the...

Let nand k be positive integers and let G be a graph. A set D of vertices of G is defined to be an (n, k )-dominating set of G if every vertex of V( G) D is within distance n from at least k vertices of D. The minimum cardinality among all (n, k )-dominating sets of G is called the (n, k )-domination number of G and is denoted by 'Yn,k(G). A set I of vertices of G is defined to be an (n, k)inde...

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...

We consider the dominating point set reporting problem in two-dimension. We propose a data structure for finding the set of dominating points inside a given orthogonal query rectangle. Given a set of n points in the plane, it supports 4-sided queries in O(log n+k), where k is size of the output, using O(n log n) space. This work can be of application when range queries are generated using mobil...

Let k be a positive integer. A vertex subset D of a graph G = (V,E) is a perfect k-dominating set of G if every vertex v of G, not in D, is adjacent to exactly k vertices of D. The minimum cardinality of a perfect k-dominating set of G is the perfect k-domination number γkp(G). In this paper, we generalize perfect domination to perfect k-domination, where many bounds of γkp(G) are obtained. We ...

Let G be a simple graph without isolated vertices with vertex set V (G) and edge set E(G) and let k be a positive integer. A function f : E(G) → {−1, 1} is said to be a signed star k-dominating function on G if ∑ e∈E(v) f(e) ≥ k for every vertex v of G, where E(v) = {uv ∈ E(G) | u ∈ N(v)}. A set {f1, f2, . . . , fd} of signed star k-dominating functions on G with the property that ∑d i=1 fi(e) ...