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 <inf>u</inf> ⊆ N(u) ∩ (V − D) such that |Cu|≤⌈d(u)k⌉ and ⋃ <inf>u ∈ D</inf> C <inf>u</inf> = 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 γdk(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. © Springer Nature Switzerland AG 2019.