Bounds on k-part degree restricted domination number of a graph

Abstract

For a positive integer k, a dominating set D of a graph G is said to be a k-part degree reslrictedm dominating set (k-DRD set) if for all u 2 D, there exists a set [Formula Presented] The minimum cardinality of a k-DRD set of a graph G is called the k-part degree restricted domination number of G and is denoted by ?<inf>d/k</inf> (G). In this paper, we provide some bounds on ?<inf>d/k</inf> of join of two graphs, bounds on ?<inf>d/k</inf> in terms of maximum degree, independence number and covering number. Further, we discuss some Nordhaus-Gaddum type results. In addition to this, we prove that for any graph G, ?<inf>d/k</inf> (G)??<inf>k</inf> (G), where ?<inf>k</inf> (G) is the k-domination number of G and we characterize the trees T for which ?<inf>d/k</inf>(?)= ?<inf>k</inf>(G). © 2021, Tsing Hua University. All rights reserved.