On the distance domination number of bipartite graphs

Abstract

Let G be a graph and k be a positive integer. A vertex set D is called a k-distance dominating set of G if each vertex of G is either in D or at a maximum distance k from some vertex of D. k-distance domination number of G is the minimum cardinality among all k-distance dominating sets of G. In this note we give upper bounds on the k-distance domination number of a connected bipartite graph, and improve some results have been given like Theorems 2.1 and 2.7 in [Tian and Xu, A note on distance domination of graphs,

An algebraic approach to sets defining minimal dominating sets of regular graphs

“…In order to show Condition (ii), we should check that for each r − 1 = 3 distinct integers 1 ≤ j 1 , j 2 , j 3 ≤ 6, there exists a positive integer 1 ≤ i ≤ 11 such that a i,j 1 + a i,j 2 + a i,j 3 = 0. To do this, we have to examine it 6 3 = 20 times. This proves that M (S) satisfies Condition (ii).…”

“…, 11. Hence, we have to check Condition (iii) exactly 6 4 + 6 5 + 6 6 = 22 times. After checking them, we derive that Condition (iii) holds.…”

“…In this section, we state the definitions which we will use in the rest of this paper. For any unexplained notation and terminology, we refer the reader to [1,2,5,6,12,13].…”

A survey on enhanced power graphs of finite groups

“…The concept of domination is very important in graph theory. Recently, it has been considered, for example, in [13,18,37,42,45,58,68]. It is obvious that the identity element e is a dominating vertex of every enhanced power graph P e (G).…”

Connected domination value in graphs

“…1. Introduction 1 The study of dominating sets, domination number and other variants of domination parameters of a graph like [1,3,4,5,6,11,13] forms an integral part of both theoretical as well as practical aspects of graph theory. However, a systematic local study of domination has not been studied extensively.…”