Some New Perspectives on Global Domination in Graphs

A dominating set is called a global dominating set if it is a dominating set of a graph and its complement . Here we explore the possibility to relate the domination number of graph and the global domination number of the larger graph obtained from by means of various graph operations. In this paper we consider the following problem: Does the global domination number remain invariant under any graph operations? We present an affirmative answer to this problem and establish several results. 1. Introduction The domination in graphs is one of the concepts in graph theory which has attracted many researchers to work on it. Many variants of dominating sets are available in the existing literature. This paper is focused on global domination in graphs. We begin with simple, finite, and undirected graph with . The set is called a dominating set if . A dominating set is called a minimal dominating set (MDS) if no proper subset of is a dominating set. The minimum cardinality of a dominating set in is called the domination number of denoted by , and the corresponding dominating set is called a -set of . The complement of is the graph with vertex set in which two vertices are adjacent in if and only if they are not neighbors in . A dominating set of is called a global dominating set if it is also a dominating set of . The global domination number is the minimum cardinality of a global dominating set of . The concept of global domination in a graph was introduced by Sampathkumar [1]. This concept is remained in focus of many researchers. For example, the global domination number of boolean function graph is discussed by Janakiraman et al. [2]. The NP completeness of global domination problems is discussed by Carrington [3] and by Carrington and Brigham [4]. The global domination number for the larger graphs obtained from the given graph is discussed by Vaidya and Pandit [5] while Kulli and Janakiram [6] have introduced the concept of total global dominating sets. The discussion on global domination in graphs of small diameters is carried out by Gangadharappa and Desai [7]. The wheel is defined to be the join where . The vertex corresponding to is known as apex vertex, and the vertices corresponding to cycle are known as rim vertices. Duplication of an edge of a graph produces a new graph by adding an edge such that and . The shadow graph of a connected graph is constructed by taking two copies of , say and . Join each vertex in to the neighbors of the corresponding vertex in . A vertex switching of a graph is the graph obtained by taking a vertex of , removing all