Clustering is an important task in the process of data analysis which can be viewed as a data modeling technique that provides an attractive mechanism to automatically find the hidden structure of large data sets (Jain et al., 1999). Informally, this task consists of the division of data items (objects, instances, etc.) into groups or categories, such that all objects in the same group are simi...

Let γ(G), i(G), γS(G) and iS(G) denote the domination number, the independent domination number, the strong domination number and the independent strong domination number of a graph G, respectively. A graph G is called γi-perfect (domination perfect) if γ(H) = i(H), for every induced subgraph H of G. The classes of γγS-perfect, γSiS-perfect, iiS-perfect and γiS-perfect graphs are defined analog...

A subset S of V is called a dominating set in G if every vertex in V-S is adjacent to at least one vertex in S. A Dominating set is said to be Fuzzy Double Dominating set if every vertex in V-S is adjacent to at least two vertices in S. The minimum cardinality taken over all, the minimal double dominating set is called Fuzzy Double Domination Number and is denoted by γ fdd (G). The minimum numb...

The bondage number of a graph G is the minimum number of edges whose removal results in a graph with larger domination number.A dominating setD is called an efficient dominating set ofG if |N−[v]∩D|=1 for every vertex v ∈ V (G). In this paper we establish a tight lower bound for the bondage number of a vertex-transitive graph. We also obtain upper bounds for regular graphs by investigating the ...

Abstract— Let G be a bipartite graph. A X-dominating set D of X of G is a strong nonsplit Xdominating set of G if every vertex in X-D is X-adjacent to all other vertices in X-D. The strong nonsplit X-domination number of a graph G, denoted by ) (G snsX γ is the minimum cardinality of a strong nonsplit X-dominating set. We find the bounds for strong nonsplit X-dominating set and give its biparti...

We count the numbers of independent dominating sets of rooted labeled trees, ordinary labeled trees, and recursive trees, respectively.

We study the existence and the number of k-dominating independent sets in certain graph families. While the case k = 1 namely the case of maximal independent sets which is originated from Erdős and Moser is widely investigated, much less is known in general. In this paper we settle the question for trees and prove that the maximum number of kdominating independent sets in n-vertex graphs is bet...

We pursue the study of families of functions on the natural numbers, with emphasis here on the bounded families. The situation being more complicated than the unbounded case, we attack the problem by classifying the families according to their bounding and dominating numbers, the traditional scheme for gaps. Many open questions remain.

Let G = (V, E) be a graph without isolated vertices. A secure edge dominating set of G is an edge dominating set F⊆E with the property that for each e ∈ E – F, there exists f∈F adjacent to e such that (F – {f}) ∪ {e} is an edge dominating set. The secure edge domination number γ's(G) of G is the minimum cardinality of a secure edge dominating set of G. In this paper, we initiate a study of the ...