Relation Between k-DRD and Dominating Set

Abstract

In this paper, a new parameter on domination is defined by imposing a restriction on the degrees of vertices in the dominating set. For a positive integer k, a dominating set D of a graph G is said to be a k-part degree restricted dominating set (k-DRD-set), if for all u ∈ D there exists a set C _u ⊆ N(u) ∩ (V − D) such that $|C_u| \leq \lceil \frac {d(u)}{k}\rceil$ and ⋃_{u ∈ D} C _u = V − D. The minimum cardinality of a k-part degree restricted dominating set of G is called the k-part degree restricted domination number of G and is denoted by $\gamma _{\frac {d}{k}}(G)$. Here, we determine the k-part degree restricted domination number of some well-known graphs, relation between dominating and k-DRD set, and an algorithm which verifies whether a given dominating set is a k-DRD set or not.